Efficiency Evaluation through Radial and Non

Efficiency Evaluation through Radial and NonRadial Measures in DEA Networking Processes Thesis submitted to Pondicherry University in partial fulfillment of the requirements for the award of the degree of

DOCTOR OF PHILOSOPHY IN

STATISTICS BY

Arif Muhammad Tali Under the Guidance and Supervision of

Dr. P. Tirupathi Rao Associate Professor & Head

Department of Statistics Ramanujan School of Mathematical Sciences Pondicherry University (A Central University) Puducherry-605014, India November 2017

PONDICHERRY UNIVERSITY (A Central University)

DR.P.TIRUPATHI RAO Associate Professor& Head PU/STAT/2017-18/

DEPARTMENT OF STATISTICS Ramanujan School of Mathematical Sciences R. V. Nagar, Kalapet, Puducherry-605014,India Email:[email protected],[email protected] Phone: 91-413-2654-390 &829 (Office), 91–9486492241 (Hand Phone) Dated: 20-11-2017

CERTIFICATE This is to certify that the work incorporated in this thesis entitled “Efficiency Evaluation through Radial and Non-Radial Measures in DEA Networking Processes” submitted to Pondicherry University in partial fulfillment of the requirement for the award of degree of Doctor of Philosophy in Statistics, is a bonafide record of research work carried out by Mr. Arif Muhammad Tali under my guidance and supervision and no part of this thesis has been submitted for the award of any Degree / Diploma / Associateship / Fellowship or any similar titles before.

Place: Puducherry

(Dr. P. Tirupathi Rao)

Date:

Research Supervisor and Guide

i

I dedicate this Thesis to my parents,

Gh Muhammad and Aisha Banoo

& Brothers Sayar Ahmad and Irfan Ahmad

ii

DECLARATION BY THE CANDIDATE

I hereby declare that the work incorporated in this thesis entitled “Efficiency Evaluation through Radial and Non-Radial Measures in DEA Networking Processes” submitted to Pondicherry University in partial fulfillment of the requirement for the award of degree of Doctor of Philosophy in Statistics, is a record of original research work carried out by me under the supervision of Dr. P. Tirupathi Rao, Associate Professor & Head, Department of Statistics, Pondicherry University and that no part of this thesis has been submitted to any Degree / Diploma / Associateship / Fellowship or any similar titles before in this University or elsewhere.

Place: Puducherry Date:

(Arif Muhammad Tali) Ph.D. Research Scholar Department of Statistics, Pondicherry University, Puducherry-605014, INDIA

iii

Acknowledgement Praise and thanks be to Allah, First and last, Lord and Cherisher of all words who inspires entire humanity towards knowledge, truth and eternal commendation, who blessed me with strength and required passionate ardor to overcome all the obstacles in the way of this toilsome journey. In utter gratitude I bow my head before HIM. This thesis is the result of HIS abundant blessings on me in infinite ways. I am ever grateful to the Almighty for HIS untold providential care. Words are not sufficient for expressing my intensity of sentiments. So, most humbly I express my deep sense of gratitude to my reverent supervisor Dr. P. Tirupathi Rao, Associate professor & Head, Department of Statistics, Pondicherry University, whose constructive

criticism,

affectionate

attitude,

strong

motivation

and

persistent

encouragement have always been a source of inspiration. I am grateful to him for unsparingly helping me by sharing the in-depth knowledge of the subject, which has significantly improved this research. Apart from the subject of my research, I learnt a lot from him, which I am sure, will be helpful in different stages of my life. I feel very fortunate to work under his able supervision and I thank him sincerely from the depth of my heart. Besides my supervisor, I offer my sincere thanks to Dr. Kiruthika, Department of Statistics

and

Dr.

K.

Chandrasekara

Rao,

Department

of

Banking

Technology,

Pondicherry University for serving on my doctorial committee and for their guidance, insightful comments, and suggestions. I

express

my

Mathematical

deepest

sciences,

gratitude

Pondicherry

to

Prof.

P.

University

Dhanavanthan, for

his

Dean

valuable

School

advice

of

and

encouragement. I wish to extend my sincere gratitude to all the faculty members of the department Dr. J. Subramani, Dr. Navin Chandra, Dr. Sudesh Pundir, Dr. R. Vishnu Vardhan and V. S. Vaidyanathan for their moral support and wholehearted cooperation. I also pay my sincere thanks to all non-teaching staff of the department for their kind help and cooperation I wish to thank Prof. Aquil Ahmad, Department of Statistics & Operations Research, Aligarh Muslim University, Dr. M. A. k. Baig,

Dr. Tariq Rashid Jan and Dr. Sheikh

Parvaiz, Department of Statistics, University of Kashmir for their encouragement time by time during the study period. I am greatly indebted to all my research colleagues and friends who encouraged me throughout this work. I am especially grateful to Mr. Qaiser Farooq, Sharon Varghese, Vikash Kumar, N. Padmapriya, R. Deepana, M. Ajiths, H. Rehman, Dr. Mashroor Ahmad

iv

and Dr. Kiran Kumar for creating a healthy environment and sharing of ideas during the research. It is also my bounden duty to express my hearty thanks to my friends Dr. Aasif Shah, Dr. Abdul Gafar, Manzoor Hassan, Mir Mehrajudin

Mudasir Bashir, Zahoorul Haq,

Showket Bashir, Shariq Ahmad, Showket Ahmad, Ishfaq Ahmad, Rameez Raja, , Munirul Islam, Younis Ahmad, Sakib Hassan , Rouf Bhat and all others for their good wishes and constructive help from time to time. Words fail me to express my indebtedness and gratitude to my parents, Gh. Muhammad and Aisha Banoo for their unequivocal support throughout, as always, for which my mere expression of thanks likewise does not suffice. They have done everything possible to see me at this place. They have always helped me in good and bad times alike to keep me focused towards my goal. Their unflagging love, energetic support and persistent confidence in me, has taken the load off my shoulder and clearing the path towards thesis completion. I feel shortage of words to pay thanks to them. I would also like to express my special thanks to my brothers Sayar Ahmad and Irfan Ahmad and all other relatives who devoted their trust and love to me throughout my years of studies. I take this opportunity to express my gracious concern towards all those who have helped in various ways throughout the development stages of this thesis. I wish to express my apology for not being able to mention all the names in this small space. I am very much thankful to DST-SERB India for providing me travel support to attend an international Conference at Prague, Czech Republic Lastly, I would like to acknowledge University Grant Commission for providing me the ‘National Fellowship for OBC students’ which was a great economic booster for me. Finally, for any errors or inadequacies that may remain in this work, of course, the responsibility is entirely my own.

Arif Muhammad Tali

v

Acronyms and Abbreviations BCC : Banker Charnes Cooper CCR : Charnes Cooper Rhodes CCR-I: Input Oriented CCR-O: Input Oriented CMIE : Centre for Monitoring Indian Economy CRS : Constant Returns-to-Scale CV : Coefficient of Variation DEA : Data Envelopment Analysis DFA : Deterministic Frontier Analysis DMU : Decision Making Unit DRS : Decreasing Returns-to-Scale DSS : Decision Supporting System EE : Economic Efficiency FDEA : Fuzzy Data Envelopment Analysis FDH: Free Disposal Hull FPP: Fractional Programming Problems GDP: Gross Domestic Product IO: Input-oriented IRS: Increasing Returns-to-Scale LDM: Least-Distance Measure LFP: Linear Fractional Programming LPP: Linear Programming Problem MLE: Maximum Likelihood Estimation MPI: Malmquist Productivity Index MPSS: Most Productive Scale Size vi

NDEA: Networking Data Envelopment Analysis NPDF: Non-Parametric Deterministic Frontier NPFA: Non-Parametric Frontier Analysis NPSF: Non-Parametric Stochastic Frontier PC: Productivity Change PPS: Production Possibility Set RBI: Reserve Bank of India R&D: Research and Development RAM: Ram Adjusted Model RTS: Returns-to-scale SBM: Slack-Based Measure SE: Scale Efficiency SFA: Stochastic Frontier Analysis TC : Technological Change TEC: Technical Efficiency Change TFP: Total Factor Productivity VRS: Variable Returns-to-Scale

vii

NOTATIONS J, (j=1,...,n) = The set of n DMUs under study jo : DMU under evaluation m = Total Number of Inputs m(q) = Total Number of Inputs at stage ’q’ s = The Number of Outputs s(q) = The Number of Outputs ar stage ’q’ D = The number Intermediates w : denotes the window width P: The production possibility set ξ: Very small positive number xij , (i = 1, ..., m): ith input of the j th DMU (q)

xij , (i = 1, ..., m): ith input of the j th DMU at stage ’q’ zdj , (d = 1, ..., D): dth intermediate of the j th DMU (q)

zdj , (d = 1, ..., D): dth intermediate of the j th DMU at stage ’q’ yrj , (r = 1, ..., s): rth output of the j th DMU (q)

yrj , (r = 1, ..., s): rth output of the j th DMU at stage ’q’ Λj : Weight of j th DMU in fractional problem λj : Weight of j th DMU in linear form (q)

λj : Weight of j th DMU at stage ’q’ th s− Input Slack i : i th s+ Output Slack r : r −,(q)

: ith Input Slack at stage ’q’

+,(q)

: rth Output Slack at stage ’q’

si

sr

Ej = Efficiency score in multiplier models θj : Input oriented efficiency score of the j th DMU

φj : Output Oriented efficiency score of the j th DMU ρj : SBM efficiency score of the j th DMU ρj : SBM efficiency score of the j th DMU ρqj : SBM efficiency score of the j th DMU at stage ’q’ ρj : Optimistic SBM efficiency score of the j th DMU φj : Pessimistic SBM efficiency score of the j th DMU etwork : Optimistic overall network SBM efficiency score of the j th DMU ρN j etwork φN : Pessimistic overall network SBM efficiency score of the j th DMU j

ξjN etwork : Network efficiency based on double frontiers S t : Production technology at time ’t’ Dt (xt , y t ): Distance function with reference technology ’t’ Dt+1 (xt , y t ): Distance function with reference technology ’t+1’ Mjt : Malmquist Index of DMU j with reference technology t Mjt+1 : Malmquist Index of DMU j with reference technology t+1 SEjt : Scale efficiency change at time t.

ix

List of Figures 1.1

Production Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Methods of Efficiency Evaluation . . . . . . . . . . . . . . . . . . . .

4

1.3

Regression Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

The Deterministic Production Frontier . . . . . . . . . . . . . . . . .

6

1.5

The Stochastic Production Frontier . . . . . . . . . . . . . . . . . . .

7

1.6

FDH Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.7

CCR Production Frontier . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8

Scale and Pure Technical Efficiency . . . . . . . . . . . . . . . . . . . 17

1.9

Non-radial Efficiency Approach . . . . . . . . . . . . . . . . . . . . . 19

1.10 Distribution of DEA-related articles by year (1978-2016) . . . . . . . 26 2.1

Type-I of series type two-stage DEA processes . . . . . . . . . . . . . 43

2.2

Type-II of series type two-stage DEA processes . . . . . . . . . . . . 48

2.3

Type-III of series type two-stage DEA processes . . . . . . . . . . . . 51

2.4

Illustration on Type-III of two-stage processes . . . . . . . . . . . . . 53

2.5

Type-IV of series type two-stage DEA processes . . . . . . . . . . . . 55

2.6

Multi-stage series production process . . . . . . . . . . . . . . . . . . 59

3.1

Optimistic and Pessimistic DEA Frontiers . . . . . . . . . . . . . . . 65

3.2

DEA Double Frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . 66

x

3.3

Two-Stage DEA Structure . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1

Cement Production of India . . . . . . . . . . . . . . . . . . . . . . . 87

4.2

Efficiency Trend over windows . . . . . . . . . . . . . . . . . . . . . . 99

4.3

Comparison of Two Approaches . . . . . . . . . . . . . . . . . . . . . 100

5.1

Malmquist Productivity Index . . . . . . . . . . . . . . . . . . . . . . 118

5.2

Productivity, Efficiency and technological changes . . . . . . . . . . . 124

xi

List of Tables 1.3.1 Envelopment Form of DEA

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2.3.1 Factors used in efficiency evaluation . . . . . . . . . . . . . . . . . . . 46 2.3.2 Results based on Type-I production process . . . . . . . . . . . . . . 47 2.3.3 Results based on type-III process . . . . . . . . . . . . . . . . . . . . 54 3.5.1 Efficiency Scores and Ranking Based on Double Frontiers . . . . . . . 82 4.5.1 Windows Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5.2 Descriptive Statistics of Variables . . . . . . . . . . . . . . . . . . . . 95 4.5.3 Window Analysis of SBM Efficiency Scores . . . . . . . . . . . . . . . 96 4.5.4 Window Analysis of SBM Efficiency Scores . . . . . . . . . . . . . . . 97 4.5.5 Average SBM Scores through Different Methods . . . . . . . . . . . . 99 5.3.1 CCR-Output Oriented Efficiency during 2012-16 . . . . . . . . . . . . 114 5.3.2 Super Efficiency of Banks during 2012-16 . . . . . . . . . . . . . . . . 115 5.3.3 CCR and BCC efficiency scores based on year 2016 . . . . . . . . . . 116 5.4.1 MPI changes during 2012-16 & Productive changes over years . . . . 125 5.4.2 Efficiency change & Technical change over years . . . . . . . . . . . . 126

xii

ABSTRACT The present study has made an attempt to focus on some contributions to DEA in terms of both classical and empirical perspectives. In the former context, we proposed some theoretical models to be employed in the resembled environment with defined assumptions whereas in later context, certain models were employed to the real data-sets and explored the possible inferences related to performance and efficiency analysis. Conventional DEA models consider the process as blackbox. In contrast, two-stage DEA paradigm considers the internal; flow within the process and identifies the sources of inefficiency. In particular, attempt has been made to formulate the SBM models for four possible types of series type two-stage DEA processes with respective real-world illustration. Additionally, we formulated SBM models for two-stage process in double frontier case. The thesis has further extended the scope of DEA window analysis that examines the productivity trend of all DMUs over time. Furthermore, DEA-based MPI has been considered to Productivity change, efficiency change and technological change over time. We have observed few things. First, network models are more reliable when the production process exhibits in several stages. Second, the double frontier based performance evaluation is more feasible if the conditions of the production processes are complex to identify. Third, window analysis is important tool to examine the fluctuation in the performance overtime assuming constant technology. Finally, in terms of MPI, we evaluated the data set of Indian commercial banks and observed that total productivity changes of DMUs were influenced heavily by technical change rather than efficiency change. Keywords: DEA, Efficiency Evaluation, Two-stage DEA, DEA Window Analysis, DEA-based MPI

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Contents DECLARATION

iii

ACKNOWLEDGEMENTS

iv

ABBREVIATION

vi

NOTATIONS

viii

List of Figures

ix

List of Tables

xii

ABSTRACT

xiii

1 General Introduction 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Measurement of Efficiency . . . . . . . . . . . . . . . . 1.2.1 Ratio Analysis . . . . . . . . . . . . . . . . . . 1.2.2 Regression Analysis . . . . . . . . . . . . . . . . 1.2.3 Frontier Analysis . . . . . . . . . . . . . . . . . 1.3 Data Envelopment Analysis (DEA) . . . . . . . . . . . 1.3.1 Charnes Cooper and Rhodes (CCR) Model . . . 1.3.2 The Banker Charnes Cooper (BCC) Model . . 1.4 Slack Based Model (SBM) . . . . . . . . . . . . . . . . 1.5 Network DEA Processes . . . . . . . . . . . . . . . . . 1.6 Double Frontier DEA . . . . . . . . . . . . . . . . . . . 1.7 DEA Window Analysis . . . . . . . . . . . . . . . . . . 1.8 DEA-Malmquist Productivity Index (DEA-MPI) . . . . 1.9 Literature Review . . . . . . . . . . . . . . . . . . . . 1.9.1 Literature Review on Theoretical Developments 1.9.2 Literature Review on Empirical Contributions . 1.10 RESEARCH GAPS AND MOTIVATION . . . . . . . 1.11 OBJECTIVES OF STUDY . . . . . . . . . . . . . . . 1.12 Data Acquisition and Processing Methodology . . . . . xiv

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1 1 3 3 4 5 9 11 15 18 21 22 23 24 25 25 30 32 33 34

1.13 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 34 2 Two-Stage DEA Production Processes 2.1 Introduction . . . . . . . . . . . . . . . . . . . . 2.2 Slack-Based Measure of Efficiency . . . . . . . . 2.3 SBM Models in Two-Stage Processes . . . . . . 2.3.1 Type-I Structure of Two-stage Process . 2.3.2 Type-II Structure of Two-stage Process . 2.3.3 Type-III Structure of Two-stage Process 2.3.4 Type-IV Structure of Two-stage Process 2.4 Multi-Stage Series Process . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . .

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3 SBM Double Frontiers in Two-Stage Production Processes 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 DEA Double Frontiers . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Radial Models for DEA Double Frontiers . . . . . . . . 3.2.2 Non-Radial SBM Models for DEA Double Frontiers . 3.3 SBM Double Frontiers for Two-Stage DEA Processes . . . . . 3.3.1 Double Frontiers for the First Stage . . . . . . . . . . . 3.3.2 Double Frontiers for the Second Stage . . . . . . . . . 3.3.3 DEA Double Frontiers for the Network System . . . . . 3.4 Overall Efficiency Based on Double Frontiers . . . . . . . . . . 3.5 Numerical Illustration. . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Multi-Period Efficiency Evaluation Through Average SBM 4.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2 DEA in Indian Cement Industry . . . . . . . . 4.3 Methodology DEA Window Analysis . . . . . 4.4 The Approach of Average SBM . . . . . . . . 4.5 Data and Selection of Variable . . . . . . . . . 4.5.1 Window Analysis . . . . . . . . . . . . 4.5.2 Average SBM Measure . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . .

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62 62 65 66 69 72 73 75 77 80 80 82

Window Analysis & . . . . . . . .

5 Multi-Period Performance Evaluation of Indian Through DEA and MPI 5.1 Introduction . . . . . . . . . . . . . . . . . . . . 5.2 Reported Research Work in DEA with MPI . . 5.3 Data Envelopment Analysis and its Testing . . . 5.3.1 DEA Concept . . . . . . . . . . . . . . . 5.3.2 Charnes Cooper Rhodes (CCR) Model . xv

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36 36 38 42 42 47 50 55 58 60

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84 84 87 88 91 93 95 99 101

Commercial Banks 102 . . . . . . . . . . . . 102 . . . . . . . . . . . . 104 . . . . . . . . . . . . 107 . . . . . . . . . . . . 107 . . . . . . . . . . . . 108

5.3.3 Banker Charnes Cooper (CCR) Model . . . . 5.3.4 Scale Efficiency . . . . . . . . . . . . . . . . . 5.3.5 DEA Model for Super-Efficiency . . . . . . . 5.3.6 Variables and Analysis . . . . . . . . . . . . . 5.3.7 Data Collection and Processing Methodology . Malmquist Productivity Index (MPI) . . . . . . . . . 5.4.1 Analysis of Malmquist Productivity Index . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . .

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109 111 112 112 113 117 123 127

6 Summary, Findings and Scope for Future Research 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.2 Summary and Contribution . . . . . . . . . . . . . . 6.3 Findings and Observations of the study . . . . . . . . 6.4 Scope for Future Research . . . . . . . . . . . . . . .

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128 128 129 132 135

5.4 5.5

Bibliography

137

Appendix

152

xvi

Chapter 1

General Introduction 1.1

Introduction

Decision making is one of the core areas of science to have the wise and effective administration of any entity. In order to draw valid decisions across profit and non-profit organizations, several methods have been proposed by various decision makers to help managers to remain alive in the cut throat competition. One of such methods is the performance valuation and benchmarking which is used most often to identify and adopt best practices to enhance the performance and productivity of a entity.This evaluation consists of collecting, analysing and reporting information regarding the performance of any entity or a Decision Making Unit (DMU). A DMU is distinguished as technically efficient, if it is not possible to decrease/increase an input/ output without worsening other corresponding inputs/ outputs, otherwise, DMU would be distinguished as inefficient. However, the linear combination of efficient DMUs generates a frontier for all other inefficient DMUs that can be obtained through the corresponding production functions such as Linear, Quadratic, Cobb-Douglas, Trans log etc. The production function will reveal the technical way of producing outputs from given inputs through production frontier. Additionally, it will generate different output levels for a fixed technology. However, the inefficient DMUs which fall below the frontier can reach frontier through two different ways. One by enhancing the output level with fixed input level until it will reach the frontier, referred as output orientation. The second way is to fix the 1

output level and minimize the input level, referred as input orientation. by joining the points corresponding to maximum possible outputs from a certain level of inputs called as output orientation and similarly by joining the points corresponding to minimum inputs to produce a certain level of outputs, called as input orientation. Thus, input and output orientations are two possible way for the inefficient units to reach the frontier where they are labeled as efficient. Furthermore, if the inputs and outputs are fully disposable then there is possibility of mixed orientation where both inputs are reduced and outputs are increased simultaneously. The production function for a single input-output case is depicted in Figure 1.1.

Figure 1.1: Production Frontier In the Figure 1.1, the technically efficient points such as A, B, and C are positioned on the production frontier, while technically inefficient point D lies under the frontier. The point D can reach the frontier in two different ways. One way is to reduce input from X2 to X0 while producing same output level Y0 so as to reach point A in input orientation case. The second way called as output orientation case, output must be increased from Y0 to Y2 from the existing input level x2 . As mentioned above, If the inputs and outputs are strongly disposable the projection can go anywhere between A and C. If both input and output are optimized by the same proportion till it will reach to the frontier, then in that case it will reach to point B. In case, if the inputs and outputs are optimized by different proportions, it will fall on the frontier anywhere 2

between A and C. The figure only depicts the direction rather than the amount. The amount by which the inputs and outputs are to optimized can be obtained by the different methods of efficiency evaluation.

1.2

Measurement of Efficiency

It is important to present the main approaches of performance evaluation as they reveal the foundation for the methodological framework. There are various studies related to estimating efficiency through some mathematical models. The term efficiency as a definition and measurement was first introduced by Koopmans et al. (1951). Some Distance functions were implemented by Debreu (1951) for output expansion directions in multi-output case, along with radial distance measurement from the production frontier. Similarly, Shephard (1953) implemented distance functions for contraction of inputs. However, the overall functionality of these measures was never realized, resulted in the suggestions of some parametric and non-parametric methods (Farrell, 1957). Some developments were made to these contrasted models by many authors. All those models can be broadly classified into three different categories. Figure 1.2 shows the broad categorization of all efficiency evaluation methods.

1.2.1

Ratio Analysis

This approach is considered to be easiest among all approaches for evaluating the measure of performance by using different indicators as ratios. The effective utilization of particular input reveals its efficiency, which can be profit per employee, output per labor, turnover ratio etc. These ratios provide a partial measure of efficiency and sometimes provide misleading results (Sherman, 1984). It is also difficult to rank DMUs on the basis of these partial ratios in multi-input multioutput case, as one ratio will better for one DMU and other for second DMU. However, the overall measure of efficiency can be measured by calculating a number of ratios simultaneously.

3

Figure 1.2: Methods of Efficiency Evaluation

1.2.2

Regression Analysis

Regression analysis deals with the exploration of association between two variables that is dependent variable (output) and some independent variables (inputs). The relationships between these two variables are usually represented by a fixed structural forms such as multiple linear regression forms (Simar and Wilson, 2000), whose estimation in our context aims to identifying the efficiency. Figure 1.3 depicts an arbitrary example of single input and single output case of linear regression approach. This method examines the average expected quantity of output for each quantity of inputs used. The figure represents the fitted curve through average efficiency value. The DMUs are compared with respect to obtained average line. The main advantage of regression analysis over the ratio analysis is that it can accommodate multiple inputs responsible for the production of particular output, which is not possible in case of Ratio Analysis. Furthermore, it requires a certain production function which reveals the relationship of an output with different inputs. However, the problem with the regression analysis is that it cannot include more than

4

Figure 1.3: Regression Analyses one output in a single investigation. For multi-output case, a series of investigations can be done to evaluate the performance of each output. The most important limitation of the regression analysis is that it evaluates the average efficiency rather the maximum efficiency. Here, DMUs are to be compared with average efficiency. There is no meaning of comparison for more efficient DMUs.

1.2.3

Frontier Analysis

Frontier Analysis is the approach which can adapt several inputs and several outputs in a single measurement. Furthermore, frontier analysis evaluates the maximum efficiency rather than average efficiency. Farrell (1957) suggested this approach of measuring the efficiency. Frontier analysis approach can be broadly classified into two main approaches such as (1.2.3.1) Parametric Frontier Approach and (1.2.3.2) Non-Parametric Frontier Approach. Both these approaches evaluate the efficiency with respect to the best frontier. 1.2.3.1

Parametric Frontier Analysis (PFA)

The Parametric Frontier Analysis is one of the approaches of frontier analysis, which requires a prior specific form of production form. In PFA, the production function could be Linear, Quadratic, Trans log or Cob-Douglas. The parameters of the corresponding function can be obtained either from econometric regression techniques or through mathematical programming problems (Jacobs, 2001). The parametric approach consists of two sub-approaches aims to estimate the coefficients 5

of the production function either based on the deterministic parametric frontier or stochastic parametric frontier. These two sub-approaches are briefly described as (i) Deterministic Frontier Analysis and (ii) Stochastic Frontier Analysis. (i) Deterministic Frontier Analysis (DFA): This method assumes deterministic relationship between output and inputs in order to reach a DMU to a production frontier. As mentioned above, it is necessary to have predefined specific production function. The inputs and outputs represent the independent and dependent variables respectively. Deviation (if any) from the frontier for a particular DMU is considered only because of technical inefficiency. Therefore, production function assumed to be fully deterministic according to Smith and Street (2005). The methods developed for estimating the parameters of the function are ordinary least square by conventional authors and mathematical programming by Aigner and Chu (1968). DFA has both advantages and disadvantages over other approaches, in the former case; it does not require any distributional properties of inefficiency,whereas in the latter case, it neglects the random errors which may be a possible cause for deviation from the frontier.

Figure 1.4: The Deterministic Production Frontier Figure 1.4 represents the DFA, where the units [A, B, C] are technically efficient as they lie on the production frontier and unit D under the frontier implies technically inefficient unit, whose inefficiency amounts to be line CD. This deviation is only because of technical efficiency as random errors are not possible in this case. The parametric approach consists of two sub-approaches aims in estimating all coefficients of the production function either based on the deterministic parametric frontier or stochastic parametric frontier. These two sub-approaches are briefly described in the following sub-sections. 6

(ii) Stochastic Frontier Analysis (SFA): This method was developed by Aigner et al. (1977) based on the idea of decomposing the total cause of deviation from the frontier into two complimentary events due to technical inefficiency and due to uncontrollable random errors. The main advantage of SFA is that we can deal separately with the component of technical inefficiency. On the other hand, this method requires certain specific distributional form of both the components. The commonly used distributions for the said analysis include normal, half-normal or gamma (Smith and Street, 2005).

Figure 1.5: The Stochastic Production Frontier Figure 1.5 represents the SFA frontier curve where unit D is technically inefficient and its deviation from the frontier is not only because of technical inefficiency but also due to some random shocks. The line segment DC can be separated into two segments where DE represents technical inefficiency and EC represents random errors. 1.2.3.2

Non-Parametric Frontier Analysis (NPFA)

NPFA method is easier and useful than others as it does not require any parametric production function. This method neither needs any production function nor accompanied of parameters. NPFA methods considers the the non-parametric approach especially Programming Problems (LPPs) for evaluating the performance scores and treats any deviation from the frontier as because of technical inefficiency. In other words, in NPFA there is no need to estimate different parameters. These approaches can be further categorised into (i) Non-Parametric Deterministic Frontier and (ii) Non-Parametric Stochastic Frontier.

7

(i)Non-Parametric Deterministic Frontier: Non-Parametric Deterministic Frontier (NPDF) does not need any particular production function. It has two main properties, one it a non-parametric approach and second is that is based on deterministic frontier. NPDF has two approaches, Data Envelopment Analysis (DEA) and Free Disposal Hull(FDH). Data Envelopment Analysis is a non-parametric approach to access the relative performance of set of DMUs using several inputs to produce several outputs. This approach is based on the linear programming problems for evaluation of such scores. The approach was proposed by Charnes et al. (1978) as a measure of efficiency for not-for-profit organizations, later; it was recognized as the best tool to evaluate efficiency in other dimensions also. This technique examines the efficient DMUs among the set of DMUs that will constitute the best practice frontier and all the DMUs are evaluated with respect to evaluated frontier. This research study proposed various methods of DEA for measuring the efficiency and is elaborated in the later sections. Free Disposal Hull was introduced by Deprins et al. (2006). They considered the similar approach of DEA with a difference of relaxing the convexity assumption between the efficient units. In DEA, any point between two efficient points is also efficient, whereas it is not in the case of Free Disposal Hull Approach (FDH) approach. Hence the frontier in FDH looks like a stair type as shown in Figure 1.6 and is useful where the inputs and outputs are not fully disposable. Figure 1.6 displaying the isoquant of two inputs responsible for the production of an output. The isoquant AB represents a fixed amount of output which can be produced with different combinations of inputs. FDH frontier is a stair type as inputs cannot be fully disposable. As we will see in next section, the efficiency frontier associated with DEA method is based on convex combinations of different efficient units whereas, it is relaxed in the FDH approach. This method is important in cases of integer cases or non-disposable input cases. This approach gets higher values as compared to DEA, because FDH curve is always enveloped by DEA curve. (ii) Non-Parametric Stochastic Frontier (Stochastic DEA): It aims to overcome the disadvantages of DEA of not taking into account the inherent random errors. This method was proposed by Sengupta (1987) by considering the production function as unknown and the distribution of the output is obtained through bootstrapping. This method will generate many pseudo samples from the original set of 8

Figure 1.6: FDH Frontier observations through bootstrapping and simulation processes to get the underlying distribution fairly accurate. Hence, efficiency can be evaluated without knowing the structure of production function.

1.3

Data Envelopment Analysis (DEA)

In the above section two alternative approaches, parametric approaches, and non-parametric approaches were discussed for assessing the technical efficiency of DMUs. Preferring one on the other is not an easy task to decide as both of them having their own preferences and limitations. This study is based on the nonparametric method of operational research called as DEA. DEA is a methodology used to examine the performance of a set of DMUs that transforms set of homogeneous inputs into the set of homogeneous outputs. This non-parametric approach was developed by Charnes et al. (1978). In this approach, linear programming problems (LPPs) are constructed whose constraints give rise an empirical production zone and to assess the productive efficiency of any DMU in focus. These LLP’s are used to obtain a piece-wise non-parametric frontier, which envelops all the studied DMUs. This frontier is generated without the need, to parameterize the production function. This approach was formulated on the 9

basis of the concept of relative efficiency given by Farrell (1957). He assumed that the distance from the efficient production frontier provides the efficiency score for each DMU. However, the Farrell’s model has a limitation that it can be used only for the problems of single input and single output analysis. Charnes et al. (1978) surmounted the limitation by extending the Ferrell’s model to introduce several inputs and several outputs into the analysis. The increased capacity becomes the most attractive element of DEA. DEA involves in finding the solutions of LPPs of observed inputs and outputs for the selection of weights to get highest possible efficiency score. The solutions of these LPPs provide the relative efficiency score of the DMUs and is done in two steps. The first one consists in formulating the virtual DMU from the actual inputs and outputs of corresponding DMUs and the second includes in the determination of maximum possible inputs required to produce a certain level of output for a particular DMU. If the score comes out to be one for a DMU, then there is no other DMU in the set that can outperform this DMU, hence it referred as efficient. On the contrary, if the ratio comes out to be less than 1, then it is called as inefficient DMU because there presents a virtual DMU that can produce same output by using less fraction of inputs as compared to it. On the other hand, the DEA methodology is too accompanied with several disadvantages to other conventional methods. The primary disadvantage includes that DEA cannot comprise stochastic variables. In other words, there is no space for error term, hence taking technical inefficacy as a sole reason for any deviation from the frontier. The other disadvantage with DEA is that it treats the process of DMUs as Black-box, i.e., it neglects the internal processes which are responsible for overall process in the analysis. It considers only the two end products of the process which are inputs initially used and outputs finally produced. Another considerable disadvantage of DEA is that it is sensitive to the selection of input and output variables. Similarly, DEA obtains the efficiency based on optimal favorable cases, which means that there are always favorable conditions for DMUs. However, this is not always possible in the case of real life as there may be situations where DMUs are operating on unfavorable conditions. The vital problem with the DEA is that it cannot be used in time-varying cross-sectional data to obtain the efficiency change. In other words, a DMU efficient at particular period may or may not be efficient at some other time. One may be interested in estimating the efficiency change over time which is not possible in DEA. Furthermore, in case time-varying data there is not 10

only possibility of technical efficiency changes but also technological change.

1.3.1

Charnes Cooper and Rhodes (CCR) Model

Charnes et al. (1978) developed a model grounded on the Farrell’s concept of relative efficiency which evaluates technical efficiency of set of DMU transforming a homogeneous vector of inputs into the homogeneous vector of outputs. CCR model is based on constant returns assumption. The very reason of CCR model is to estimate the performance of each DMU relative to best set of DMUs. Suppose the set contains ’n’ DMUs where DM Uj , (j = 1, · · · , n) represents j th DMU in the set which uses ’m’ dimensional input vector to generate ’s’ dimensional output vector. Let xij , (i = 1, · · · , m) and yrj , (r = 1, · · · , s) represents respectively the input and output vector of j th DMU. The production possibility set (PPS) denoted by P is the feasible set of points defined by CCR model has the following four assumptions; (i) Each observed point xj , yj belongs to P, that is xj , yj ∈ P, ∀ j (ii) If xj , yj ∈ P , then αxj , αyj ∈ P, ∀ j where α is any positive number. (iii) If x, y ∈ P , then there exists a point x¯, y¯ with x¯ > x and y¯ < y such that x¯, y¯ ∈ P . (iv) If points A and B belong to P, then αA + (1 − α)B ∈ P , for any positive α. The four assumptions of CCR model can be summarized in the following expression given by Cooper et al. (2006). P =

n

n n o X X x, y | x ≥ λj xij , y ≤ λj yrj , λj ≥ 0 j=1

(1.3.1)

j=1

Charnes et al. (1978) on the basis of above four assumptions formulated the

11

mathematical model for any DM Ujo , (jo ∈ J) as follows; Pn j=1 ur yrjo M aximize Ejo = Pn j=1 vi xijo

(1.3.2)

Subject to Constraints; Pn ur yrjo Pj=1 ≤ 1; ∀ j = 1, · · · , n n j=1 vi xijo ur , vi ≥ ε∀r, i The optimal value of Ejo reveals the efficiency score of DMU jo and the values of ur and vi represents the respective weights of rth output and ith input respectively. The ε is a very small positive real number to ensure strict the positivity of weights of all inputs and outputs. The optimal weights are obtained by solving the corresponding LPPs. The objective function of above fractional programming problem contains the ratio of virtual outputs and virtual inputs for a particular DMU with the restriction that this ratio should be less than or equal to one for each of the DMUs. The model is solved iteratively for each DMU to evaluate their corresponding efficiency scores. As the above model is in fractional form, it can be transformed into linear one by applying the Charnes Cooper transformation. The transformed model is referred as Primal form or Multiplier form of CCR model. If the denominator is constrained to equal to one, which follows the assumption that there is no other DMU or virtual DMU that produces more output than DMU jo with all DMUs using same input level. The primal form is described as follows; M aximize Ejo =

s X

µr yrjo

r=1

Subject to Constraints; m X νi xijo = 1

(1.3.3)

i=1 s X

µr yrj −

r=1

µr , νi ≥ ε;

m X

νi xij ≤ 0, ∀ j = 1, · · · , n

i=1

∀ i, r

The orientation of the problem depends upon the condition either weighted inputs or weighted outputs are constrained to be equal to one. In the model 1.3.3, 12

the first constraint means that weighted sum of inputs is constrained to one, which makes it output orientation problem. The second constraint is to restrict all the DMUs below frontier, .i.e., their efficiency will be no more than one. The problem with this primal form is that its solutions are computationally difficult as the number of constraints directly depends on the number n of DMUs. On the other hand, the dual or envelopment form has relatively very less constraints as the constraints of dual models depend on a number of inputs and outputs (m + s) rather than number of DMUs. It is held that number of DMUs must be thrice as compared to number of input and outputs combined to get the fair results. which are always less than the number of DMUs. Thus, it is feasible to deal with dual form rather than primal form. Furthermore, while formulating the dual model it incorporated the additional slack variables which represent the possible input excesses and output shortfalls. The envelopment form of CCR model will be; M inimize θjo − ε

X m i=1

s− i

+

s X

s+ r

r=1

Subject to Constraints; n X λj xij + s− i = θjo xijo ;

(1.3.4) i = 1, · · · , m

j=1 n X

λj yrj − s+ r = yrjo ;

r = 1, · · · , s

j=1 + λj , s− i , sr ≥ 0, ∀ j, i, r

The model 1.3.4 represent the envelopment form of input-oriented model with the CRS assumptions. The aims of this model is to minimize the current input level while fixing certain output level. Similarly, the output orientated model can be formulated to increase the output with a certain input length. In the model, s− i + and sr are the input excesses and output shortfalls respectively. The variable λj represent weight for j th DMU and ε is very small positive quantity. Here θjo is the efficiency score with the condition 0 ≤ θjo ≤ 1 . If it is equal to one then DMU is operating on the frontier. In other words, there is no other DMU or a virtual DMU which surpasses its production with same inputs or same out with relatively less input level. The model is operating in such a way that the efficiency score is equal to one if and only if slacks are zero. On the other hand, if it is operating under

13

frontier then efficiency score will be less than one with positive slack values. For such DMUs the model provides peer units or reference units (refjo ) which will be obtained from the equation as; ∗ refjo = j/λj ≥ 0 ; j = 1, · · · , n

(1.3.5)

These reference DMUs are the role models for inefficient DMUs to follow their way to become efficient. A DMU which lies under the frontier have specific targets to reach the frontier. A measurements of these input and output targets (˜ x, y˜) to reach the frontier and become efficient can be obtained from the following relations. x˜ijo = θ∗ xijo − s−∗ i y˜rjo = yrjo + s+∗ r

(1.3.6)

For inefficient DMU, x˜ij ≤ xij and y˜rj ≥ yij . Thus, in order to be efficient, they have to make inputs and outputs as per the targeted values. Following Figure 1.7 represents the CCR model.

Figure 1.7: CCR Production Frontier Figure 1.7 is the simple example of one input and one output perspective with CRS assumption. Due to this assumption, only DMU at point A is CCR efficient because of its efficiency score θjo is equal to one, while all other DMUs (B, C, D, E) lying below frontier and hence their efficiency will be smaller than one (< 1). There 14

is no point or any linear combination of points which outperforms the performance of point A.

1.3.2

The Banker Charnes Cooper (BCC) Model

As mentioned, CCR model is established on the assumption of CRS which means that any proportional change of inputs results in a similar change of outputs. For example, if it is possible to produce ’Y’ from ’X’, then under CRS assumption 0 αY 0 should be produced from 0 αX 0 , where α is any positive constant. Hence, it implies that size is neglected in the CRS case. However, the efficiency of a DMU is very much influenced by its size. For example in banks, the size of assets, deposits, loans etc, may influence the banking operations. Thus, CRS assumption seems highly unrealistic. The efficiency estimates suffer if the scale of DMU is neglected. To relax the CRS assumptions Banker et al. (1984) proposed an extended model to adopt variable returns to scale (VRS) assumption and named it as BCC model. The VRS assumption consists all three possibilities of returns to scale. Thus, it contains increasing returns and decreasing returns apart from constant returns. The production possibility set for BCC model is comprised as; P =

n

x, y | x ≥

n X

λj xij , y ≤

j=1

n X

λj yrj ,

j=1

n X

λj = 1, λj ≥ 0,

o

(1.3.7)

j=1

The BCC model could be formulated by adding the unconstrained scalar 0 ω 0 in the obejective function of CCR model Banker et al. (1984); M aximize Ejo =

s X

µr yrjo + ω

r=1


(1.3.8)

i=1 s X

µr yrj −

r=1

µr , νi ≥ ε;

m X

νi xij + ω ≤ 0, ∀ j = 1, · · · , n

i=1

∀ i, r;

ω is unrestricted.

Here the variable ω may be positive, negative or zero. If it is equal to zero,

15

then it is same as CCR model. Furthermore, ω being free ensures that the frontier consists of several combinations of best practices. Here the frontier comprises of three different parts. First is with increasing returns to scale (IRS) where ω ≤ 0 the second is with constant returns to scale (CRS) where ω = 0 and the third is with decreasing returns to scale (DRS) where ω ≥ 0 . It gives the opportunity to compare the DMUs according to their sizes. In other words, the BCC model does not neglect the size of DMU but estimates its influence on the efficiency. The dual model in case of BCC model looks like as follows; M inimize θjo − ε

X m i=1

s− i

+

s X

s+ r

r=1

Subject to Constraints; n X λj xij + s− i = θjo xijo ; j=1 n X j=1 n X


(1.3.8) i = 1, · · · , m r = 1, · · · , s

λj = 1

j=1 + λj , s− i , sr ≥ 0, ∀ j, i, r

The primary difference between the BCC model in equation 1.3.8 and the CCR model 1.3.4 is that of convexity constraint. The scalar θjo represents the proportional reduction of all inputs simultaneously to be fully efficient. A DMU is fully efficient if θjo = 1 with all slacks as zero, otherwise, if θjo < 1 and/or slacks are non-zero then it is inefficient. Similarly, output oriented DEA model can be formulated. Here the Table 1.3.1 presents the both input and output oriented models in the envelopment form. The BCC model compares DMUs according to their sizes as small size DMUs are compared with small one, as all of them belongs to IRS while as large-sized are accordingly compared with the large DMUs as they all belong to DRS. This can be shown in Figure 1.8. Figure 1.8 reveals that the DMU positioned at point ’A’ comes out to be CCRefficient as well as BCC-efficient. CCR curve is comprised of scale effect while as it is excluded in the BCC curve. Thus, the difference between the curves represent 16

Table 1.3.1: Envelopment Form of DEA Frontier Type M in. θjo

Input oriented Pm − Ps + − ε s + s i=1 i r=1 r

M ax. φjo

Output oriented Pm − Ps + + ε s + s i=1 i r=1 r

Subject to Constraints; Subject to Constraints; Pn Pn λj xij + s− = θjo xijo ; i = 1, · · · , m j=1 λj xij + s− i i = xijo ; i = 1, · · · , m j=1 Pn Pn CRS + + j=1 λj yrj − sr = yrjo ; r = 1, · · · , s j=1 λj yrj − sr = φjo yrjo ; r = 1, · · · , s − + − + λj , si , sr ≥ 0, ∀ j, i, r λj , si , sr ≥ 0, ∀ j, i, r Pn Add VRS j=1 λj = 1 NIRS

Add

Pn

λj ≤ 1

NDRS

Add

Pn

λj ≥ 1

j=1

j=1

Figure 1.8: Scale and Pure Technical Efficiency the scale effect. The area between the CCR and BCC curves represents the scale inefficiency. Mathematically scale efficiency can be measured as; Scale Efficiency =

T ECRS T EV RS

The technical efficiency of inefficient DMU C is calculated to segment CQ and 0 CQ’ to the BCC and CCR model respectively. As CQ > CQ, the CCR model has basically overvalued the technical efficiency of DMU C. The radial measure of 17

0

the segment CQ represents the pure technical inefficiency and the segment QQ 0 represents the scale inefficiency. If DMU C reduces input level from x to x , it will 00 become BCC efficient and if x to x then it will become CCR efficient in input orientation case. . A similar analysis can be arbitrarily done in output orientation case. In order to get rid of orientation non-radial SBM model have been proposed.

1.4

Slack Based Model (SBM)

The models discussed in all the above sections provide a radial measure of efficiency. These radial measures provide a degree or a proportion by which all the inputs/outputs must be simultaneously decreased/increased depends upon the orientation of the problem. For example, if the technical efficiency for a particular DMU in input orientation case comes out to be 0.90, then according to usual DEA models, all the inputs need to be reduced by 10 percent. But in real life, all the inputs cannot be reduced simultaneously, for example, the area under particular factory cannot be reduced, but labor and operating expenses can be reduced. However, in some cases, some inputs can be reduced to more extent than others, in that situations non-radial models can be applied. Moreover, sometimes radial models will provide weak efficient, whereas it will not in the case of non-radial models. Non-radial DEA models are the efficiency assessment approaches that do not depend on the simultaneous reduction of all inputs or simultaneous extension of all outputs. Here some inputs can be reduced to some more extent than others or some outputs can be more extended than others. The difference between radial and non-radial measures can be shown in the Figure 1.9. Figure 1.9 represents the example of one input and two outputs case, where DMUs (A, B, D) are technically efficient, while DMUs (C, F,G, H, I) are technically inefficient. These inefficient DMUs can be efficient if they increase both the outputs by the same proportion. For example, the point H is inefficient, but if it increases 0 both the outputs by the same proportion until it will reach the frontier at H . This assessment of efficiency is called radial accumulation. Similarly, F will reach to 0 0 0 F by increasing both the outputs by the same proportion. Here F and H are 0 dummy efficient units. On comparing ’A’ with F it be observed the DMU ’A’ 0 produces more of output-1 than F with same level of second output. Thus, there is a possibility for ’F’ to further increase the output-1 which can be attained by

18

Figure 1.9: Non-radial Efficiency Approach 0

lateral moment from F to A. Since this achievement can be made without the worsening the other output. Now the proportions of two outputs will not remain same, hence called as scale efficiency. Thus for DMU F to be efficient, it has to reach F’ in the first stage to become radial efficient and then lateral movement from F to A in the second stage to become scale efficiently. This radial efficiency and scale efficiency can be obtained in two phases of DEA. In the first phase, radial efficiency is obtained, then in the second stage substituting radial efficiency in the second phase to obtain the scale efficiency. However, a unified approach has been done where both the efficiency scores are obtained simultaneously in a single model, this model is called SBM model, was given by Tone (2001). This model directly goes F to A by increasing both the outputs by different proportions. Suppose we have n DMUs to be examined, each one using m inputs to produce s outputs, with the PPS 1.3.1. We consider an expression for a particular DMU from the reference set as; xj o =

n X

λj xij + s− i

j=1

y jo =

n X

λj yrj − s+ r

(1.4.1)

j=1 − + m + s With λj , s− i , sr ≥ 0, and si ∈ < , sr ∈ < are respectively the vectors of input excesses and output shortfalls which are together referred as slacks. Hence, the following expression will satisfy the property of unit variance and monotone.

19

M inimize τj0 =

1−

1 m

1+

1 s

Si− i=1 xij0

Pm

Sr+ r=1 yrj0

Ps

Subject to Constraints; n X Λj xij + Si− = xij0 ; i = 1 , 2 , 3 , . . . m.

(1.4.2)

j=1 n X

Λj yrj − Sr+ = yrj0 ; r = 1 , 2 , 3 , . . . s.

j=1

Λj ≥ 0; Si− ≥ 0; Sr+ ≥ 0, ∀j, i, r. This is a non-linear (Fractional) programming problem and can be transformed into linear one by applying the Charnes-Cooper transformation. The transformed model in linear form is given as; m

1 X s− i M inimize ρj0 = t − m i=1 xij0 Subject to Constraints; s 1 X s+ r t+ =1 s r=1 yrj0 n X j=1 n X

(1.4.3)

λj xij + s− i = txij0 ; i = 1 , 2 , 3 , . . . m. λj yrj − s+ r = tyrj0 ; r = 1 , 2 , 3 , . . . s.

j=1 + λj ≥ 0; s− i ≥ 0; sr ≥ 0, ∀j, i, r; and t > 0.

This slack-based model can measure the non-radial efficiency estimator. Here, if ρ∗ = 1, then the corresponding DMU is efficient and all slacks will be equal to zero, otherwise, it is inefficient.

20

1.5

Network DEA Processes

In conventional1 DEA models and their extensions, production process are considered to be black-boxes where the internal sub-processes are neglected in the analysis process. In such cases, the initial inputs and final outputs of the process are only considered for analysis while ignoring the inter-mediating stages. There are numerous real-life problems need to be evaluated having several inter-stages within the process. Thus, applying conventional models to such type of processes seems to be absurd. This perspective is often inappropriate and insufficient; as such approach provides no insight regarding the locations of inefficiency and will not able to improve the efficiency. Thus to get insights in the process we have to see what is happening in the black box. The network DEA paradigm is the multistage process which examines the flow of production through several stages and accordingly helps in locating the source of inefficiency. Nishimizu and Hulten (1978) recognized the intermediate measures to study the sources of Japan economy. Färe and Grosskopf (1996) were among the first who developed the frontier model for productivity measurement with intermediate products. There are various types of network processes where the production happens to be in stages. These stages may be in any number with different connections between them. Usually, they are connected as either in parallel or in series type of processes with two, three or multi-stages. Two-stage DEA process is one of the important and most used processes of network DEA processes. There are number of studies based on two-stage DEA processes. In two-stage DEA problem, DMUs produced goods or services in two stages, where the first stage consists in transforming initial inputs into some outputs. The second stage, utilizes the transformed material of first stage to generate a final product which will leave the production process. The in between measures, i.e, transfromed one from first stage are termed as intermediates. For example, banks utilizes labor, capital, and assets to generate deposits which in turn are used in the second stage to generate interest income and loan recovered as the final output. Seiford and Zhu (1999) examined the efficiency of US commercial banks with two-stage process. They consider the profitability of banks by taking labor and assets as inputs to produce profits and revenue as outputs 1

These models includes the models developed by Charnes et al. (1978) and their extensions (Handbook on Data Envelopment Analysis, by Cooper et al. (2011))

21

at the first stage. The second stage utilizes the intermediates to generate market value and earnings per share as final outputs. There are cases where some external factors are influencing the process at any stage of two-stage production process. For example, in many situations, some outputs immediately leaves the system right at the first stage which are not further used for production process. In some other situations, apart from intermediates some external inputs are employed to furnish the final production in the second stage of the process. Thus, it makes four different possible structures which are eloborated in the second chapter of this thesis.

1.6

Double Frontier DEA

Conventional DEA models examines the best relative efficiency while assuming the favorable condition for the production processes. In all such types of models, the ratio of weighted sum of outputs and weighted sum of inputs is maximized and the resulting efficiency score is called best relative efficiency score or sometimes referred as optimistic relative efficiency or simply optimistic efficiency. In other words, all the conventional models assigns the most favorable weights to all DMUs through LPP’s such that the optimistic efficiency is maximized. If the optimistic relative efficiency comes out to be one for a particular DMU, it is said to be optimistic efficient DMU or DEA efficient; otherwise, if it came below one, it is referred as DEA inefficient or optimistic non-efficient. It is believed that the Optimistic non-efficient DMUs have always worst performance than optimistic efficient DMUs. This method provides a scalar measure to optimize the inputs and outputs of optimistic non-efficient DMUs. This may happen either by enhancing their current output levels to the obtained measure or by contracting their current input levels to the evaluated score depends upon the orientation of the problem. On the other hand, if the ratio of weighted outputs and weighted inputs as mentioned above, is minimized with the restriction that the same ratio is no less than one, the resulting efficiency score is called as worst relative efficiency or pessimistic efficiency. In doing so, one could obtain totally opposite frontier as obtained by optimistic DEA model, because by minimizing the problem, each DMU under evaluation will acquire the most unfavorable weights.In this approach, if efficiency score is one the DMU is called as pessimistic inefficient DMU and if it is greater

22

than one then it is called as pessimistic non-inefficient DMU. It is the fact that the DMUs on the pessimistic frontier are always worse than the DMUs under the frontier. The DMU on pessimistic frontier have more chance of being vanished off from the market. Maximizing (Optimistic) and minimizing (Pessimistic) the same LPP generates the two extreme frontiers for each DMU in the study set. If there are favorable conditions for the production processes, then optimistic frontier give better estimates. If the conditions regarding the favorability and unfavorably are unknown, then any evaluation method based on only one of them is considered to be biased. In such cases, the efficiency estimates should be based on the both optimistic and pessimistic frontiers. Any approach that determines the efficiency on both frontiers is called as double frontier DEA (Wang and Chin, 2009). A Detailed discussion and methodology will be presented in chapter three.

1.7

DEA Window Analysis

All the DEA approaches for efficiency evaluation are employed for a particular time period and cramping in case of cross-sectional and time-varying data. A DMU efficient at a particular period need not necessarily be efficient at different periods. One could be interested in estimating the amount of efficiency, changed between any two different time periods. In such cases, a DEA model is used on the repeated basis, e.g. the so-called DEA window analysis method. In simple words, it is one of the methods used to verify productive change over time. This method was proposed by Charnes et al. (1984) and is working on the principle of moving averages i.e, one year is added to analysis while other is removed. Thus, it is a technique used in observing the performance trends of a DMUs over years. Window analysis evaluates the efficiency of each window by considering each DMU as dissimilar in different time period. In dong so, it results in increasing the number of DMUs. In other words, A DMU in five years is treated as five DMUs which can be compared to each other and also with other DMUs. This way of analysis will be useful in less number of DMUs with more inputs and outputs. The analysis consists in comparing each DMU with itself in different windows and with other DMUs in the same window, results in increasing number of comparisons in problems with limited DMUs. The number of comparisons depends on the window width. The longer the window width will be the lesser the number of comparisons

23

and the lesser the window width will be more the number of comparisons. The range of window width is arbitarily choosen and could be somewhere between one and the total number of periods under study. Charnes et al. (1984) observed that window’s with width three or four are most feasible to do such type of analysis. As it is good to have small window width we have taken window width as three. Although this approach is useful in evaluating the efficiency trends at the same time, it ignores the effect of technological change over the different periods. The effect of technological change on the total productivity change is done through Malmquist productivity index which is given in next section and fully elaborated in chapter five.

1.8

DEA-Malmquist Productivity Index (DEAMPI)

DEA window analysis is used to determine the technical efficiency change over time for each DMU in the set. However, it neglects the effect of technological change over time. As over the period of time, there is a possibility of not only the technical efficiency change but also technological change. Here it is better to consider productivity rather than the efficiency as produtivity is dynamic whereas efficiency has static nature. This implies that efficiency neglects the time in production process whereas it is important factor in productivity. Thus, if there is productivity change between two periods, it indicates the a technological change as well as technical efficiency change. Thus, examining productivity looks very imperative. Index numbers are one of the methods to estimate such measures. Caves et al. (1982a) proposed the Malmquist productivity index (MPI) based on the idea of Malmquist (1953) to estimate the productivity change. They proposed the model as the ratio of two distance functions with reference technology of base year. The MPI commonly called as Total Factor Productivity (TFP) determines any progress or regress in production technology as well as evaluates the amount of change in the technical efficiency over the time. Fare et al. (1994) formulated the non-parametric based MPI to measure the productivity change through linear programming problems. Their model is based on the two reference technologies and comprises of four components which are equivalently represented by their corresponding LPPs. This model becomes the standard methodology to examine the productivity change over time and is used in several studies for DEA analysis of efficiency changes for various profit 24

and non-profit organizations.

1.9

Literature Review

The current study aims at providing some contribution in the area of DEA. Our contribution is in theoretical as well as in empirical aspects. In theoretical aspects, we developed some models for efficiency measurements for some real-life situations, while in the later case; we employed some existing models for different data sets of real organizations to draw some valid conclusions regarding the managerial performances. Hence, a reported research is reviewed according to theoretical development and empirical contributions in two separate sections.

1.9.1

Literature Review on Theoretical Developments

The idea of technical efficiency evaluation was first introduced by Farrell (1957). This pioneering model is applicable to single input/output case and failed to the measurement of multiple inputs and outputs into any satisfactory measure of efficiency. Almost after twenty years, Charnes et al. (1978) developed a model based on Farrell’s idea to assess the relative efficiency of multi-input and multi-output production processes and titled it as Data Envelopment Analysis(DEA). The basic idea behind DEA was to formulate the methodology to identify the best practice DMUs which makes efficient frontier. Furthermore, it finds the efficiency of non-frontier units and identifies benchmarks against which such inefficient DMUs can be compared. As this model was based on CRS assumption, it was further extended by Banker et al. (1984) to include VRS assumption. Since the advent of DEA in 1978, there is an impressive growth in both theoretical and applied aspects. In theoretical aspects, various models were proposed to estimate different efficiency measures wherein the latter case, the proposed models were used as a performance assessment tools in a variety of organizations like banking, education, health-care, banking, agriculture, production companies, airports and many more profit as well as non-profit organizations. For details see [Amirteimoori (2006); Chen and Zhu (2004); Jahanshahloo et al. 2004; Kao and Hwang (2008); Khalili et al. (2010); Lozano et al. (2013); Wang et al. (2011); Wu et al. (2013)]. The basic model of Wu et al. (2013) was immediately recognized as the best tool to assess the efficiency and to have effective decisions, which results in the large contribution of research interests and decision 25

makers working in this area. One can infer the importance of the area from the Figure 1.11

Figure 1.10: Distribution of DEA-related articles by year (1978-2016) As mentioned in earlier sections, radial and non-radial consists the approches of DEA models. The CCR BCC models come under the category of the radial one as they reflect the score on proportional changes. These radial measures overestimate the technical efficiency when there iare positive slacks, leading researchers to propose alternative efficiency measures that account slacks. The initial attempt was done by Färe and Lovell (1978) who introduced their model based on Russell measure in input orientation case. The problem with their model is that it minimizes input slacks only while allowing the slacks in the output constraints. Charnes et al. (1985) proposed an additive model based on both input excesses and output shortfalls but failed to provide a scalar measure of efficiency in terms of slacks. To have a scalar measure of efficiency which is based on slacks, various attempts have been don to normalize the slack variables in order get a scaler measue which repsents the efficiency and slacks simultaneously. There were some critiques of the additive model by Green et al. (1997) and Pastor et al. (1999). Later, Färe et al. (1985) modified their original input-oriented model (Färe and Lovell, 1978) and presented Russell graph measure based on non-linear programming problems. Their model was based on both input and output slacks, thus, provides a consistent measure of efficiency. There are several other models to measure scalar efficiency that incorporates all input and output slacks. Cooper et al. (1999) proposed a Range-adjusted measure 26

(RAM) of efficiency which maximizes the normalized input and output slacks. The notable contribution was done by Tone (2001) slacks-based measure (SBM), which maximizes the input and output slacks with a scalar measure of efficiency. In the current study, we have used the Tone’s SBM measure of efficiency to develop some models and obtain several efficiency measurements. Although DEA evaluates the measure of relative performance of a set of DMUs, it cannot locate the rootages of inefficiency in the processes where production exhibits in several stages. In such cases, the efficiency estimates may be erroneous. In contrast, a DEA network model allows one to look into the flow of intermediates and observe the internal structural performance of DMUs. The network DEA paradigm is the multistage process which helps in locating the inefficiency and observes the flow of intermediate measures among different internal stages. Nishimizu and Hulten (1978) recognize the intermediate measures to study the sources of Japan economy. Färe and Grosskopf (1996) were among the first who developed the frontier model for productivity measurement with intermediate products. Tone and Tsutsui (2009) proposed a slack-based model for measurement of efficiency in networks DEA problems. Rho and An (2007) formulated the extended two-stage DEA model based on slacks. In radial case, there are two approaches to deal with the efficiency assessment in two-stage series processes, one is multiplicative decomposition approach introduced by Kao and Hwang (2008) and other is the additive decomposition approach introduced by Chen et al. (2009). Liang et al. (2008) and Cook et al. (2010) employed the game theoretic concepts on the efficiency decomposition in two-stage production processes. A comprehensive categorized overview of the model for various multi-stage models was presented by Castelli et al. (2010). Li et al. (2012)) formulated multiplicative models for the two-stage production process with an external input apart from intermediates at the second stage. Zhou et al. (2013) employed the the idea of Nash bargaining game to access the efficiency decomposition in a simple two-stage production process. Kao et al. (2014) formulated models for the efficiency assessment in DEA network process in the frame of multi objective programming problems. Kao (2014) presented a exhaustive classification of network structures and models employed.Despotis et al. (2016) proposed additive and multiplicative models for four types of two-stage production processes and each process is illustrated with numerical data. The problem with the two-stage DEA models is that they assigns a large value to first stage than second stage. An et al. (2016) proposed a new model with fairness towards the divisional efficiency scores 27

that will evaluate unbiased estimates of efficiency scores. Entani et al. (2002) initially attempted to evaluate the performance of a DMU based on double frontiers. Their model provides interval efficiency based on DEA and Inverted DEA models. They failed to determine a scalar measure based on two extreme frontiers. Several attempts were made to overcome these problems [Wang et al. (2007); Wang et al. (2008); Azizi (2011); Azizi and Wang (2013)]. Paradi et al. (2004) identified the worst performers to represent the inefficient frontier and its application on credit risk evaluation. Jahanshahloo and Afzalinejad (2006) proposed the inefficient frontier model to rank the MUs relative to full in-efficient frontier. They designed the CCR, BCC and SBM models based on the inefficient frontier to evaluate worst relative efficiency. Shuai and Li (2005) developed a hybrid approach by combining worst practice DEA and rough set approach to determine the bankruptcy of firms. Through worst practice DEA, they identified worst performers and subsequently by rough set approach they predicts their failures. Wang and Luo (2006) fused the TOPSIS method with DEA and introduced two virtual DMUs namely ideal DMU and anti-ideal DMU. They presented the models to optimistic and pessimistic efficiency measures based on these virtual DMUs. Amirteimoori (2007) proposed an efficiency measure using ideal and anti-ideal indices which are formed on the basis of optimistic and pessimistic frontiers respectively. The fundamental reason of these two indices is to maximize the weighted distance function relative to optimistic and pessimistic production frontiers. Wang et al. (2007) obtained optimistic and pessimistic efficiency estimates independently and for overall efficiency measure. They have considered the geometric mean of two distinct and opposite estimates. The overall efficiency integrates both optimistic and pessimistic for each DMU and so is more comprehensive than any one of them taken individually. Liu and Chen (2009) proposed the worst practice frontier in non-radial SBM form and named it as WPFSBM (Worst practice frontier) to identifying bad performers in the most unfavorable scenarios. Azizi and Ajirlu (2011) proposed a novel pair of DEA models for evaluating efficiency in imprecise interval data. Similarly, Wang and Chin (2011) measured the optimistic and pessimistic efficiency in fuzzy environments and proposed fuzzy expected value approach to measure expected values of inputs and outputs. Azizi et al. (2015) presented the optimistic and pessimistic efficiency in SBM perspectives for evaluation of DMU under consideration with imprecise data. They evaluated SBM efficiency with respect to both efficient and inefficient frontiers. Further, geometric average 28

of the two opposite efficiency scores of same DMU is used to determine the DMU with the best performance. Roozbeh et al. (2015) employed the optimistic and pessimistic models with negative data to obtain Most Productive scale size (MPSS) DMUs. Aldamak et al. (2016) contributed non-convex optimistic and pessimistic models by applying Free Disposal Hull (FDH) technology. In the context of DEA window analysis, it has been relatively rarely employed than DEA in the literature. After the initial study of Charnes et al. (1984), there were several studies using DEA window analysis approach which are presented in the empirical part. Turning to measurement of productivity change over time, Malmquist Productivity Index is one of the popular methods. Malmquist (1953) initial work remains unnoticed and inapplicable until (Caves et al., 1982a) reintroduced it to productivity measurement and named after him. On the other hand, Farrell (1957) solved the distance functions through LPPs. Fare et al. (1994) take the opportunity to formulate DEA-based MPI by combining the efficiency measurement of Farrell (1957) with the productivity measurement of Caves et al. (1982a). Further, they decomposed the overall productivity into two mutually exclusive and exhaustive components, one of which measures the technical efficiency changes and the other frontier shift. The MPI models were further discussed by Chen and Ali (2004) to get more insights on the second component of MPI by identifying the strategy shifts of individual DMUs based upon the changes of isoquant. Pastor and Lovell (2005) formulated the model for obtaining a global MPI score which is circular and provides a single measure of productivity change. Yu (2007) proposed a merthod to decompose the total factor productivity into partial measures. Their new way of decomposition leads in isolation of of varous sources of TFR and provides better measures of overall productivity. Lo and Lu (2009) developed a model for inter-temporal efficiency change by means of SBM-based MPI to analyze Taiwanese financial holding companies with negative data set. Kao (2010) suggested a common-weight approach of DEA for the global MPI and applied it to analyze the productivity changes of Taiwan forests before and after reorganization. Pastor et al. (2011) introduced a biennial MPI possessing three important advantages over basic MPI; (i) avoids linear programming infeasibilities, (ii) allows for technical regress, and (iii) need to be recomputed when a new time period is added to the data set.

29

1.9.2

Literature Review on Empirical Contributions

This section presents the empirical contribution on two-stage problems. After the Färe and Grosskopf (1996) original contribution, a huge number of empirical studies on two-stage DEA problems have been done. Wang et al. (1997)studied the impact of information technology on the-the banking performance in two-stage processes. Seiford and Zhu (1999) proposed the model to examine the profitability and marketability of US commercial banks. Their study considered labor and assets as initial inputs of the first stage to produce profits and revenue as intermediates which are further used as inputs in the second stage to produce final outputs as market value, returns, and earnings per share. Zhu (2000) applied two-stage processes to the Fortune Global 500 companies. Sexton and Lewis (2003) applied two-stage DEA process to evaluate the performance of US Major League Baseball. Chen and Zhu (2004) and Rho and An (2007) extended the model of Wang et al. (1997) to evaluate the indirect impact of IT on firm performance in two-stage production process using input and output slacks. Kao and Hwang (2008) evaluated the efficiency of Taiwanese non-life insurance companies with a two-stage production process. In the first stage, they estimated the performance in premium acquisition while in the second stage profit generation was concentrated. The overall efficiency in two-stage models is typically based on geometric average or weighted average of the divisional efficiency scores. Kao and Hwang (2008), Liang et al. (2008), Du et al. (2011), Li et al. (2012) and many other studies measured the overall efficiency as the geometric average of the divisional efficiency scores. The studies of Chen and Zhu (2004), Rho and An (2007), Cook et al. (2010) etc., defined the overall efficiency based on the weighted average of individual stages. The approach of Window analysis is employed to examine the efficiency fluctuations over time. This approach was developed by Charnes et al. (1984) to monitor the efficiency fluatuations for cross-sectional and time-varying data sets. This methodology for determining the performance trends of a DMU over time is working on the principle of moving average (Yue, 1992). Carbone (2000) exemplified the application of DEA window analysis to determine the performance trends over the period of semiconductor manufacturing. Sueyoshi and Aoki (2001) studied the Japanese Postal service performances of various companies from 1983 to 1997 by combining Malmquist index and window analysis. Gu and Yue (2011) examined the seasonal efficiency change of listed commercial banks of China for the period of 2008 to 2010. 30

Chu and Lim (1998) evaluated the cost and profit efficiency through DEA window analysis of six Singapore listed banks during the period 1992-1996. From their analysis, it was revealed that stock prices determine profit efficiency rather than cost efficiency. Asmild et al. (2004) examined the performance of five large banks of Canada through DEA window analysis to determine the efficiency trends over the period for twenty years from 1981 to 2000. Sueyoshi et al. (2013) examined the environmental efficiency of US coal-fired power plants. To capture the frontier shift they employed DEA window analysis for period 1995-2007. Cullinane et al. (2004) evaluated the efficiency score of the worlds major container sea ports through DEA window analysis using panel data and cross-section data. Pjevˇcević et al. (2012) examined Serbia ports for the interval 2001 to 2008 through DEA window analysis. Yang and Chang (2009) employed DEA window analysis to examine the efficiency ˇ of integrated telecommunications of Taiwan for the period 2001 to 2005. Repkov´ a (2014) employed DEA window analysis on Czech commercial banks to examine the fluctuation in performance measures during the period 2003-2012. Window analysis is useful in the cases where the technology is set to be constant. If there is technological change over period then the window analysis will assign the technological effect to inefficiency measurement. To separate the technological effect from the technical efficiency Malmquist Productivity index (MPI) can be used, which was given Professor Sten Malmquist in 1953. Caves et al. (1982a) originally proposed a useful approach to estimate the productivity measurement in DEA through MPI. The MPI evaluates the productive efficiency of a DMU over time based on the technology of base period. Färe et al. (1992) combines the productivity measurement by Caves et al. (1982a) with the efficiency measurement of Farrell (1957) to construct a DEA-based MPI to measure the productivity change of several Swedish pharmacies over the period from 1980-1989. Their model decomposes the overall productivity into two mutually exclusive and exhaustive components namely efficiency change and technical change. Färe et al. (1992) used the radial measures to estimate DEA based productivity change, whereas same was done in non-radial case to examine Chinese major industries by Chen (2003). Further insights were done by Chen and Ali (2004) in the DEA-based MPI provided new insights. Later Pastor and Lovell (2005) extended it to global MPI. Ahn and Min (2014) examined the comparative efficiency scores of various international airports over the period 2006-2011 using DEA intended for dynamic benchmarking and MPI built on time series analysis. Bassem (2014) evaluated the productive change through DEA-based 31

MPI of 33 Middle East and North African micro finance institutions over the period 2006-2011. Wang (2015) proposed a generalized MCDA-DEA based on slacks to access the Sustainable Energy Index of 109 countries of the world for the period of 2005-2010 which comes out to be negative in the study period. Ahn and Min (2014)examined the comparative efficiency scores of various international airports over the period 2006-2011 using DEA intended for dynamic benchmarking and MPI built on time series analysis. Karagiannis and Lovell (2016) evaluated the productivity measurement with a single constant input. Their model shows that the Malmquist and HicksMoorsteen productivity indices coincide.

1.10

RESEARCH GAPS AND MOTIVATION

Lot of research has been carried out on DEA during the last two to three decades as revealed in Figure 1.10, signifies the importance of technique. An extensive review of the literature on theoretical and empirical studies of DEA reveals that there is considerable research scope. We have observed that the study has the following lacuna’s where the focus of attention is needed. (i) It is noticed that there is a huge amount of literature on Radial as well as on non-radial DEA models. Non-Radial DEA models such as Slack Based Measure (SBM) models have been extensively used in both conventional and networking DEA problems. It is evident that some attempt has been made for usage of Double Frontier Model in the two-stage production processes. Further, it is observed that there is no evidence in the literature regarding the usage of SBM models in two-stage structure with double frontiers. (ii) It is observed that there is good amount of research regarding various networking production processes, being evaluated through DEA. However, there is no evidence on the usage of DEA for evaluating the production processes with four possible combinations of Series Type - Two Stage production processes such as (a) all the outputs of stage-1 will be the inputs for stage-2, (b) partial outputs of stage-1 will be the inputs for stage-2, (c) all the outputs of stage-1 and some inputs from external sources will be the inputs for stage-2, (d) partial outputs of stage-1 and some inputs from external sources will be the inputs for stage-2. It is another area of motivation for carrying this study.

32

(iii) It is noticed that DEA evaluates the efficiency for a particular time period and is not applicable for panel data. An efficient DMU at a particular point of time need not be efficient throughout the period of panel time. However, the panel data can be dealt with DEA-Window Analysis. There are very few studies on DEA-Window Analysis in the Indian context. (iv) No study has been reported in the handling of abnormal panel data and the data with missing values. In order to address such types of data in DEA, it is more appropriate in considering the centralized and standard measures such as averages of inputs and outputs. (v) Malmquist productivity change can be decomposed into two mutually exclusive and exhaustive events; i) Efficiency change and ii) Technical Change. These two measures catching up of particular DMU and the innovation over time. This technique is very useful but no study has been done in the Indian context.

1.11

OBJECTIVES OF STUDY

Keeping the above research gaps, the following research objectives are identified. 1. To propose a slack-based model for a two-stage production process based on optimistic and pessimistic frontiers (Double Frontiers) and to apply it in real life situations. 2. To identify all possible Series Type - Two Stage production processes and to propose slack-based models for each of them along with its generalization to the Series Type -Multi-Stage case. 3. To propose a non-radial model to deal with cross-sectional and missing type data. Instead of dealing panel data, we will deal with the averages of all inputs and outputs and solve it by usual DEA models to get average efficiency. 4. To employ Malmquist Productivity Index for evaluating the productivity change, efficiency change and technical change of various Indian commercial banks and extend it to networking production problems. 5. To employ DEA-Window Analysis and to understand the behavior of efficiency changes over years and to compare them with Malmquist productivity index. 33

1.12

Data Acquisition and Processing Methodology

Our proposed models can be applied in all resembled production processes wherever the mentioned assumptions are valid. We considered MATLAB as it is convenient and compatible to all most all the software’s related to DEA. Herein, we solve LPP for each DMU separately. Our theoretical and empirical contributions are mostly with the contexts of Indian banking sector. As there is an availability of have authentic web servers for data acquisition related to Indian banks, we have collected the data from the following sources 1) Centre for Monitoring Indian Economy Pvt. Ltd. (CMIE), 2) Bloomberg Data Base, 3) Database on Indian Economy provided by Reserve Bank of Indian, and 4) Annual Reports Published by Banks. While getting the data outputs, we have used the software’s namely MATLAB, DEA Excel Solver, DEA Frontier, DEAP 2.1 and Frontier 4.1.

1.13

Organization of the Thesis

The entire thesis is designed in theoretical and empirical contributions to Data Envelopment Analysis, where the former consists in proposing some new models and extensions of existing ones, while the latter aims in employing proposed models to real-life examples. In theoretical aspects, we proposed non-radial SBM model for two-stage production processes with double frontiers. For the panel data, we proposed an average SBM model used to evaluate the average efficiency. Furthermore, we identified all possible types of Series Type-Two Stage production processes and its generalization along with their respective non-radial SBM models. The Ph.D. thesis consists of six chapters in which, Chapter 1 is on Introduction. In this chapter, we discussed the basic information about the background of the problem including a brief literature review. The notions of basic CCR and BCC models with input and output orientations were explained. Brief description about Multiplier and additive models were also provided. Research gap, motivation, and objectives of study are also presented. Chapter two is devoted to Networking DEA production processes, especially two-stage production processes. Herein, we identified all the four possible types 34

of Series Type Two-Stage production processes along with their respective SBM models. Numerical illustrations of some models are also given. We also extended series two-stage processes to Series Type- Multi-Stage production process along with its generalized SBM model. In chapter three we presented two extreme frontiers namely optimistic and pessimistic frontiers (Double frontiers) in radial as well as in non-radial forms. This includes the Double frontier of the two-stage production process in terms of nonradial (SBM) model. We further illustrate the double frontier with the help of areal-life example of non-life insurance companies in Taiwan. Chapter four is on some empirical contributions through Window Analysis. We considered DEA window analysis and applied to evaluate the efficiency change over years of study related to Indian cement companies. This also includes the proposed average SBM model to deal with panel data to obtain a scalar measure of efficiency over different years. Chapter five consists in applying the DEA and the Malmquist productivity index to a sample of Indian commercial banks for the year 2016 and for panel data of 2012-16 respectively. We first employed DEA on the data of banks for the year 2016 because of its recentness and then we employed DEA-based MPI to measure the productivity change over 2012 to 2016. As DEA-based MPI differentiates the total productive change into technical change and efficiency change, We evaluated all the three measures in the context of various Indian banks. The final chapter is devoted to Over all findings and observations, Summary and conclusions along with the future scope of the study. The empirical contribution consists of employing DEA window analysis to evaluate the efficiency of Indian cement industry. This analysis determines the trend of efficiency over the defined period. We also evaluated efficiency change over different years through Malmquist Productivity Index. The index can be divided into two mutually exclusive and exhaustive events namely efficiency change and technical change. We have estimated the productivity change, efficiency change and technical change of Indian commercial banks over different years.

35

Chapter 2

Two-Stage DEA Production Processes 2.1

Introduction

Window analysis Conventional DEA models and their modifications DEA models deals with evaluation of the efficiency of DMUs with single-stage production processes(Cooper et al., 2011). These models consider DMU as black-box, where the internal structure of the production process is ignored and hence it lacks in describing the germs of inefficiency. There are numerous real-life situations where production process passes through several internal stages and performance of these internal stages determine the overall performance. Any evaluation method that ignores these internal stages in performance evaluation may result in inaccurate efficiency evaluation. In contrast, network DEA models will investigate the process to identify the misallocation of inputs within the process and generate insights about the sources of inefficiency. Network DEA paradigm is the multistage process which helps in locating the inefficiency and indicates the flow of intermediate measures among the stages. Further, two-stage production process is a special case of networking processes where the production happens to be in two stages. In such processes, some initial inputs are consumed by the first stage to produce some outputs which are further used as inputs in the second stage to produce final outputs.The outputs of the first stage used in the second stage are called as intermediate measures. For 36

example, banks use labor, fixed assets and IT investment to generate deposits which are in turn used to generate profit and loans recovered. These two-stage production processes can be broadly classified based on two situations; (i) whether the internal structures are connected in series or parallel and (ii) whether there is any external effect at any of stage. This chapter has proposed SBM models for some two-stage production processes along with generalization to multi-stage processes. Nishimizu and Hulten (1978) initially developed a model for measuring productivity growth with intermediate measures. Their model was based on the assumption that the intermediate inputs must be explicitly recognized, and fails in developing the model where the prices are unavailable. Later, Färe and Grosskopf (1996) introduced the frontier model for productivity measurement with intermediate products that does not require that inputs are efficiently allocated among sectors or that prices are available. The efficiency of US commercial banks was evaluated in a two-stage process by Seiford and Zhu (1999), where the first stage consider the profitability of banks by taking labor and assets as inputs to produce profits and revenue as outputs. The second stage uses the outputs of the first stage as inputs to produce market value and earnings per share as outputs. Zhu (2000) applied the same two-stage process to the Fortune Global 500 companies. Sexton and Lewis (2003) evaluated the performance of Major League Baseball in a two-stage process. Rho and An (2007) formulated the extended two-stage DEA model based on slacks. . Kao and Hwang (2008) considered the two-stage process of Taiwanese non-life insurance companies with the premium acquisition in first stage and profit generation in the second stage. They estimated the efficiency of all DMUs by applying standard DEA model independently in each stage. In radial case, there are two approaches to deal with the efficiency assessment in two-stage series processes, one is multiplicative decomposition approach introduced by Kao and Hwang (2008) and other is the additive decomposition approach introduced by Chen et al. (2009). Liang et al. (2008) and Cook et al. (2010) employed the game theoretic concepts on the efficiency decomposition in two-stage production processes. A comprehensive categorized overview of the model for various multi-stage models was presented by Castelli et al. (2010). Li et al. (2012)) formulated multiplicative models for the two-stage production process with an external input apart from intermediates at the second stage. Zhou et al. (2013) employed the the idea of Nash bargaining game to access the efficiency decomposition in a simple two-stage production process. Kao et al. (2014) formulated models for the efficiency assessment in DEA network pro37

cess in the frame of multi objective programming problems. Kao (2014) presented a exhaustive classification of network structures and models employed. All the network models can be broadly classified into series and parallel type configurations. However, number of studies considers the mixture of series and parallel structures. The current study assumes the series relationship between the internal stages of the two-stage process. This assumption categorizes the two-stage DEA processes into four possible types. Despotis et al. (2016) proposed additive and multiplicative models for these types of two-stage production processes and each process is illustrated with numerical data. The problem with the two-stage DEA models is that they assigns a large value to first stage than second stage. An et al. (2016) proposed a new model with fairness towards the divisional efficiency scores. The primary task of their model is to evaluate the efficiency of first stage and fix it in the constraints of second stage so that second stage cannot get much influenced by first stage process. Further, they illustrated their model with the numerical example on Chinese commercial banks. This Chapter aims in formulating slack-based models for the four types of series type two-stage production processes. This Chapter is organized as follows. Section 2.2 is devoted to non-radial SBM measure of efficiency which present the general SBM model and its methodology. Section 2.3 presented the SBM models for two-stage process, where the sub-sections from 2.3.1 to 2.3.4 are devoted to formulate SBM models for four possible types of two-stage production process. Some of these process are also illustrated with numerical examples. The generalizations of these processes to multi-stage with Q different stages is presented in the section 2.4. The last section 2.5 is devoted to brief conclusion on the chapter.

2.2

Slack-Based Measure of Efficiency

All DEA models can be broadly classified into radial and non-radial models. Radial models provide a proportional score by which all inputs and outputs need to be optimized. Furthermore, radial models neglect the presence of any slacks. The CCR and BCC models with their extensions come under the category of radial models. On the other hand, non-radial models are based on input and output slacks while releasing the condition of proportionality; in other words, in non-radial models, inputs/outputs are allowed to decrease/increase by different proportions. Charnes

38

et al. (1985) developed additive DEA model which deals with the slacks but fails in providing an overall scalar measure. The works of Russell (1985), , Lovell and Pastor (1995), Cooper and Tone (1997) etc. were on the non radial but all were observed with some limitations. A new non-radial model namely Slack-Based Model (SBM) was proposed by Tone (2001). This model deals directly with the input excesses and output shortfalls and integrates them in the efficiency measure. In recent times, SBM measure has widely used to evaluate the efficiency scores of various production processes. Let us consider a set of ’n’ DMUs where each DM U j , (j = 1, · · · , n) uses ’m’ inputs xij , (i = 1, · · · , n) to produce ’s’ outputs yrj , (r = 1, · · · , s). The production possibility set (P) is given as; P =

n

n n o X X x, y | x ≥ Λj xij , y ≤ Λj yrj , Λj ≥ 0 j=1

(2.2.1)

j=1

A DM Ujo with (xjo , yjo ) can be described as follows; xj o =

n X

Λj xij + Si−

j=1

yjo =

n X

Λj yrj − Sr+

(2.2.2)

j=1

With Λj , Si− , Sr+ ≥ 0. The variables Si− , Sr+ are the input excesses and output shortfalls respectively , commonly called as slacks. The variable Λj is the weight assigned to j th DMU. Considering the expression 2.2.2, Tone (2001) introduced the following Fractional Programming Problem (FPP) to estimate the efficiency of a DMU.

39

M inimize τj0 =

1−

1 m

1+

1 s

Si− i=1 xij0

Pm

Sr+ r=1 yrj0

Ps

Subject to Constraints; n X Λj xij + Si− = xij0 ; i = 1 , 2 , 3 , . . . m.

(2.2.3)

j=1 n X

Λj yrj − Sr+ = yrj0 ; r = 1 , 2 , 3 , . . . s.

j=1

Λj ≥ 0; Si− ≥ 0; Sr+ ≥ 0, ∀j, i, r. As the model 2.2.3 is FPP and can be transformed to LPP by applying Charnes Cooper transformation (See Appendix-II). Let us multiply a scalar t to numerator and denominator which will not change . The scalar t should be adjusted in such a way that the denominator becomes one and can be moved to set of constraints. Thus, we have the model as m

1 X s− i M inimize ρj0 = t − m i=1 xij0 Subject to Constraints; s 1 X s+ r t+ =1 s r=1 yrj0 n X j=1 n X

λj xij + s− i = txij0 ; i = 1 , 2 , 3 , . . . m.

(2.2.4)

λj yrj − s+ r = tyrj0 ; r = 1 , 2 , 3 , . . . s.

j=1 + λj ≥ 0; s− i ≥ 0; sr ≥ 0, ∀j, i, r; and t > 0.

Let the optimal solution of the above model is as; +∗ ρ∗ , τ ∗ , λ∗ , s−∗ i , sr

(2.2.5)

Then the solution to the original SBM will be as;

40

+∗ +∗ τ ∗ = ρ∗ ; Λ∗ = λ∗ /t; Si−∗ = s−∗ i /t; Sr = sr /t

(2.2.6)

Based on the optimal solution, one can determine a DMU as efficient or inefficient. Definition 1: A DM Ujo is said to be efficient if and only if τjo = ρjo = 1. If the condition is true then this is equivalent to Si−∗ = 0 and Sr+∗ = 0 which means that there are no input slack as well as no output slack. For an SBM inefficient DMU (xjo , yjo ) we have; xj o = y jo =

n X j=1 n X

Λ∗j xij + Si−∗

(2.2.7)

Λ∗j yrj − Sr+∗

j=1

The DMU (xjo , yjo ) can approach one if it tries to minimize the input excesses and output shortfalls. Thus, SBM projection will be as follows; xjo ← xjo − S −∗

(2.2.8)

yjo ← yjo + S +∗ The inefficient DMUs need to follow their respective peer DMUs to become efficient. The peer set or reference set for an inefficient DMU (xjo , yjo ) can be obtained based on λ∗ . Definition 2: A set Ro of DMUs whose corresponding λ∗ > 0 is a reference set for an inefficient DMU (xjo , yjo ). Thus, reference set is; Rjo = j | λ∗j > 0 j = 1, · · · , n

(2.2.9)

Using the reference set the DMU (xjo , yjo ) can be represented as;

41

xj o = yjo =

n X j=1 n X

λ∗j xij + s−∗ i

(2.2.10)

λ∗j yrj − s+∗ r

j=1

Thus, the expression 2.2.9 reveals that the efficiency score depends only on the reference set of DMUs; it is not affected by the values attributed to non-reference units.

2.3

SBM Models in Two-Stage Processes

As stated earlier, in the two-stage production process, production occurs in two stages. In the first stage, initial inputs are used to produce outputs which may either leave the system or can be further used as inputs for the second stage, called as intermediate measures. If the intermediate measures are completely employed in the second stage and no other external inputs come to the second stage then it is called closed system. In contrast, if some intermediates leave the system immediately at first stage or some external inputs are used along with intermediates at the second stage then it is called open system. The intermediate products can either be in series type of production process or in parallel type of production process. Our study is based on the assumption of series type relationship between the stages. In a two-stage process, we can identify four possible types of production processes from both open and closed systems. Despotis et al. (2016) proposed radial measures for all the four types of processes in the radial case. Following their study, we proposed related non-radial SBM models for all the four categories.

2.3.1

Type-I Structure of Two-stage Process

This is the basic two-stage structure given by Kao and Hwang (2008) resembles with number of real-life examples. Herein, the production occurs in two stages and the intermediate measures are completely utilized in the second stage, hence it is closed type production system. This type of process can be illustrated through the graphical view in Figure 2.1.

42

Figure 2.1: Type-I of series type two-stage DEA processes In this elementary case (Type-I), each DMU transforms initial inputs (X) into final outputs (Y) via some intermediate measure (Z) in a two-stage process. In this type, nothing but original inputs to the first stage gets in the system and nothing but finished outputs of the second stage gets out from the system. A DMU thus could be efficient only if it is efficient inboth divisional stages. Thus, it is necessary to evaluate the efficiency of DMUs at both the stages as well as at the whole network system. We will evaluate the efficiency of individual stages by non-radial SBM measure of efficiency as given by Tone (2001). The non-radial efficiency of a DM Ujo for the first stage of type-I can be obtained through the following SBM model; m

(1)

M inimize ρjo = t −

1 X s− i m i=1 xijo

Subject to Constraints;

(2.3.1)

D

1 X s+ d =1 t+ D d=1 zdjo n X j=1 n X

λj xij + s− i = txijo

i = 1, ..., m

λj zdj − s+ d = tzdjo

d = 1, ..., D

j=1 + λj , s− i , sd ≥ 0; ∀j, i, d and t > 0

Where xijo and zdjo are respectively the inputs and outputs of the DMU under + evaluation. The variables s− i , (i = 1, ..., m) and sd , (d = 1, ..., D) are respectively the input excesses and output shortfalls of the concerned and are referred as slacks. The second stage uses nothing but ’z’ inputs (outputs from the first stage) to produce final ’s’ outputs. The SBM model for the second stage of the type-I of two-stage process is formulated by similar procedure but using different input and output 43

quantities. It is also important to mention that the weight used in the first stage cannot be same as in the second stage; hence we put different weights to inputs and outputs in the second stage. The SBM model for second stage is as follows: D

M inimize

(2) ρ jo

1 X s− d = t− D d=1 zdjo

Subject to Constraints; s 1 X s+ r t+ =1 s r=1 yrjo n X j=1 n X

(2.3.2)

µj zdj + s− d = tzdjo

d = 1, ..., D

µj yrj − s+ r = tyrjo

r = 1, ..., s

j=1 + λj , s− d , sr ≥ 0; ∀j, d, r and t > 0

Models 2.3.1 and 2.3.2 represent the SBM models to evaluate the non-radial SBM efficiency for stage one and stage two respectively. In order to formulate the model for the system (Network), it is necessary to connect the above two models. Since the outputs of first stage are the inputs of the second stage, so these two quantities must be equal and hence the following constraint guarantees the continuity of two-stages; n X j=1

λj zdj =

n X

µj zdj , d = 1, ..., D

(2.3.3)

j=1

Adding this connectivity constraint to the SBM model of the overall process will give the efficiency of the system without neglecting the effect of intermediate measures. Here, we considered the initial inputs of first stage and final external outputs of the second stage as inputs and outputs respectively. The SBM model for the network system is as follows;

44

m

M inimize

etwork ρN jo

1 X s− i = t− m i=1 xijo

Subject to Constraints; s 1 X s+ r t+ =1 s r=1 yrjo n X j=1 n X j=1 n X j=1

(2.3.4)

λj xij + s− i = txijo ; i = 1, ..., m µj yrj − s+ r = tyrjo ; r = 1, ..., s λj zdj =

n X

µj zdj ; d = 1, ..., D

j=1

λj , µj , s− , s+ ≥ 0; ∀j, i, r and t > 0 This model gives the network efficiency for the concerned DMUs. If ρ∗jo = 1 then it is network efficient, otherwise it is inefficient. A DMU may be efficient if and only if it is efficient in both the stages. If a unit comes out as inefficient, one directly evaluate the stage efficiency to locate the possible cause, which is not possible in case of non-network models. Thus, opening the black-box results in not only locate the cause for inefficiency but also helps to gain in efficiency. This model is solved for all DMUs under study, to find their corresponding system efficiency. To support this model, we tried to illustrate it with the help of real-life example taken from Khodakarami et al. (2015). 2.3.1.1

Numerical Illustration on Type-I

For illustration purpose, we apply models of first, second and network stages to the data set originally presented in Khodakarami et al. (2015). The case study concerns with the performance measurement of 27 Iranian companies (DMUs) producing resin. Table 2.3.1 presents the variables of Iranian Resin producing companies in the context of two-stage process. In the first stage, authors stressed on supply process whereas in the second stage they focused on the manufacturing process. The inputs at first stage consists of annual cost, annual personnel turnover , and environmental 45

Table 2.3.1: Factors used in efficiency evaluation Factor Notation X1j X2j Inputs X3j Z1j Intermediates Z2j Y1j Y2j Outputs Y3j

Definition Annual cost. Annual personnel turnover. Environmental cost. Number of products from supplier to manufacturer. Partnership cost in green production plan. Number of trained personnel in the fields of job, safety and health. Number of green products. Revenue.

cost . Intermediate inputs/outputs are considered as partnership cost in green production plans, and a number of products from supplier to the manufacturer. The outputs of manufacturer stage are the number of trained personnel in the fields of job, safety, and health, number of green products and revenue. Table 2.3.2 exhibits the results obtained by applying the proposed approach. Specifically, column 2 and 3 present the independent efficiency scores of stage one and two respectively, whereas the third column represents the network efficiency scores of the system. We also evaluated the efficiency score assuming black-box of the process. In other words, the last column of the table is based on the efficiency evaluation while neglecting the internal structure of the process. The performance measures of the company Aria Resin Co. for stage one, stage two and system is calculated by solving their corresponding models as given above. The results reveal that its efficiency scores at first stage, second stage and the system are 0.564, 0.374 and 0.552 respectively. These efficiency scores indicate that the mentioned is not efficient at any stage. However, the conventional measure comes out to be little greater as 0.7336. If a DMU is not efficient in any one of its stages, then it cannot be overall efficient. Thus, to become efficient DMU Aria Resin Co. has to reduce the slacks of all inputs and outputs at both divisional stages. The company Peik Chimie Co. is the only DMU which is efficient at both stages and hence network efficient. Thus, it does not have any slacks. The company (Peka Chimie Co.) is efficient at stage one but inefficient at second stage, thus, cannot be network efficient. In contrast, it is efficient if we neglect the internal structure of the process as shown in the last column. This is the property of network processes as DMU may increase their profit or decrease cost in the second stage. It is observed all companies except the company Peik Chimie Co. are either inefficient in stage one or stage two or in both, leads to inefficient system efficiency.

46

Table 2.3.2: Results based on Type-I production process Company Name (DMUs) 1. Aria Resin Co 2. Azar Resin Co 3. Peka Chemie Co. 4. Bonyan Kala Chemie Co 5. Pars Pamchal Chemical Co 6. Paint Sahar Co. 7. Taba Coatings 8. Paksan Co. 9. Chemical Carbon Acid Co 10. Alborz Chelic Co 11. Mobin Petrochemical Co. 12. Marun Petrochemical Co. 13. Fajr Petrochemical Co 14. Laleh Petrochemical Co. 15. Khosh & Kcc Co. 16. Rang Afarin Co. 17. Dorsa Chemie Co 18.BushehrChemical Industries Co. 19.Rang Avar Paint & Chemical Co. 20. Rangsazi Iran Co. 21. Petromad Kimia Co. 22. Pars Zinc Dust Co. 23. Peik Chimie Co. 24. Resin Fam Co. 25. Doreen Chimie Co 26. Pars Eshen Co. 27. Nikoo Resin Co.

ρ1∗ j 0.564 0.453 1.000 0.748 0.378 0.453 0.487 0.639 0.453 0.375 0.435 0.456 0.594 0.544 0.404 0.516 0.683 0.499 1.000 0.310 0.347 0.605 1.000 0.540 0.517 0.381 1.000

ρ2∗ j 0.374 0.471 0.466 0.805 1.000 0.695 1.000 0.377 0.466 1.000 0.553 0.931 0.261 0.474 0.416 0.378 0.776 1.000 0.677 0.248 1.000 0.736 1.000 0.323 0.317 0.489 0.370

etwork∗ ρN j 0.552 0.753 0.869 0.748 0.693 0.718 0.395 0.299 0.253 0.316 0.602 0.553 0.195 0.306 0.218 0.248 0.610 0.319 0.267 0.110 0.239 0.482 1.000 0.207 0.209 0.230 0.395

ρ∗j 0.734 0.773 1.000 0.972 0.833 0.888 0.888 0.869 0.960 0.760 1.000 0.925 0.724 0.764 0.786 1.000 0.876 1.000 0.500 0.922 0.949 1.000 1.000 0.781 0.648 1.000 0.734

Thus, to cover the distance from the current level of the criteria from the optimal level it is necessary to eliminate the slacks.

2.3.2

Type-II Structure of Two-stage Process

This production process consists of two types of outputs in the first stage, one leaves the system called as external outputs and other is used as inputs for the second stage called as intermediates. This type comes under the category of open system processes as some outputs leave the system immediately after the first stage. Herein, both stages consists external outputs. The process can be clearly understood by the graphical presentation and is given in Figure 2.2. 47

Figure 2.2: Type-II of series type two-stage DEA processes Here ’m’ initial inputs are used to produce two types of outputs, one leaves the system at first stage (external outputs of stage one) and another type is used for further process at the second stage (intermediate measures). Here, ’ m’ inputs are used to produce s(1) external outputs and ’D’ intermediate measures at first stage which are further used to produce s(2) outputs at second stage. The non-radial SBM models for individual stages and network process are as follows; m

M inimize

(1) ρ jo


Subject to Constraints; " # D s(1) +,(1) X 1 1 X s+ s r d + t+ =1 (1) D d=1 zdjo s(1) r=1 yrj o n X i = 1, ..., m λj xij + s− i = txijo ; j=1 n X j=1 n X

λj zdj − s+ d = tzdjo ; (1)

(2.3.5)

d = 1, ..., D (1)

λj yrj − sr+,(1) = tyrjo ;

r = 1, ..., s(1)

j=1 + +,(1) λj , s− ≥ 0; ∀j, i, d, r and t > 0 i , sd , sr

The expression consists of two output constraints correspond to two types of outputs. The objective function and the input constraint will remain same as previous model because there is no other input level in the system other than first stage. Similarly, we can formulate the SBM model for second stage (with weight µ), where we have inputs and outputs just like as second stage of Type-I production process.

48

The non-radial SBM model for second stage of type-II process is as follows; D

M inimize

(2) ρ jo

1 X s− d = t− D d=1 zdjo


(2.3.6)

(2)

s +,(2) 1 X sr t + (2) =1 (2) s r=1 yrj o n X µj zdj + s− d = tzdjo j=1 n X

(2)

d = 1, ..., D (2)

µj yrj − sr+,(2) = tyrjo

r = 1, ..., s(2)

j=1 + λj , s− d , sr ≥ 0; ∀j, d, r and t > 0

Models 2.3.5 and 2.3.6 evaluate respectively the SBM efficiency scores for the divisional stages first and second of type-II production process. Now what we need is the network model which will include the internal processes also. In order to connect the above models we should know the relationship between these two stages. The output of the first stage which is used as input for second stage should have similar weights at two ends. Thus, the inclusion of the constraint set Pn Pn j=1 µj zdj , d = 1, ..., D assumes the continuity of the two stages. j=1 λj zdj = P P Thus, there are nj=1 λj zdj = nj=1 µj zdj intermediate measures which are used as inputs in the stage-2. While evaluating the system efficiency it is necessary to consider the external outputs of the stage-1. These outputs may have great impact on overall efficiency; hence ignoring these outputs would lead to fallacious results. P (1) +,(1) (1) The constraints nj=1 λj yrj − sr = tyrjo ; r = 1, ..., s(1) will influence the impact of external outputs of stage-1 on the system efficiency. So, overall model for the network process of this type is as follows:

49

m

M inimize

etwork ρN jo


Subject to Constraints; # " s(1) +,(2) s(2) +,(2) 1 X sr 1 X sr + (2) =1 t + (1) (1) (2) s r=1 yrj s r=1 yrj o o n X λj xij + s− i = txijo ; i = 1, ..., m j=1 n X j=1 n X j=1 n X j=1

(1)

(1)

(2)

(2)

(2.3.7)

λj yrj − s+,(1) = tyrjo ; r = 1, ..., s(1) r λj yrj − sr+,(2) = tyrjo ; r = 1, ..., s(2) λj zdj =

n X

µj zdj ; d = 1, ..., D

j=1

λj , µj , s− , s+ ≥ 0; ∀j, i, r and t > 0 This is the required network model for the overall stage which considers the effect of final output in the first stage. There are various numerical examples which resembles with this type of production process. For example, productions of spare parts of cars are either send to market or can be sent to another production company where cars are manufactured. Likewise, there are so many problems of these types but unfortunately no data is available for analysis.

2.3.3

Type-III Structure of Two-stage Process

In type-III of two-stage series type production process, original inputs are transformed into outputs via the intermediates along with some external inputs at the second stage. In other words, intermediate measures are not enough to make production process at the second stage, it need some external inputs apart from intermediates for production process of final outputs. As the system contains two types of inputs different superscripts have been assigned accordingly. The structure of this type of production process is given in the Figure 2.3.

50

Figure 2.3: Type-III of series type two-stage DEA processes In this type of two-stage production process, m(1) initial inputs are used to produce ’D’ intermediates which are further used as inputs along with m(2) external inputs at second stage to produce final outputs. There are so many real life situations resembles with this type of production process, but the problem is the access and availability of the data. For this type of production process we will illustrate with the help of real life example of some published studies. Now, the usual SBM model for first stage of this type which is similar with the first stage of type-I, is as follows; (1)

M inimize

(1) ρ jo

m −,(1) 1 X si = t − (1) m i=1 x(1) ijo


(2.3.8)

D

t+ n X j=1 n X

1 X s+ d =1 D d=1 zdjo (1)

−,(1)

λj xij + si

(1)

= txijo ;

λj zdj − s+ d = tzdjo ;

i = 1, ..., m(1)

d = 1, ..., D

j=1 −,(1)

λj , si

, s+ d , ≥ 0; ∀j, i, d and t > 0

Now for second stage, there are external inputs along with intermediate measures for the production of final outputs. For this stage, there are 0 D0 intermediates measures and m(2) external inputs to produce final ’s’ outputs. Thus, the model will consist of two input constraints and one output constraint. Here intermediate (2) and external inputs for joth DMU are denoted by zdjo and xijo respectively. The SBM model for this type of structure is given as;

51

(2)

"

M inimize

(2) ρ jo

D m −,(2) 1 X s− 1 X si d = t− + D d=1 zdjo m(2) i=1 x(2) ijo

#

Subject to Constraints; s 1 X s+ r t+ =1 s r=1 yrjo n X j=1 n X j=1 n X

(2.3.9)

µj zdj + s− d = tzdjo ; −,(2)

(2)

µj xij + si

d = 1, ..., D (2)

= txijo ;

µj yrj − s+ r = tyrjo ;

i = 1, ..., m(2) r = 1, ..., s

j=1 − + µj , s− d , si , sr ≥ 0; ∀j, d, i, r and t > 0

Now to connect first stage and second stage, we have to add connecting constraints as like in other types of models to formulate the overall efficiency. For intermediate measures we must have; n X

λj zdj =

j=1

n X

µj zdj ; d = 1, · · · , D

(2.3.10)

j=1

For considering the effect of external inputs in the second stage we will put one more constraint like as follows; n X

(2)

−,(2)

µj xij + si

(2)

= txijo ;

i = 1, ..., m(2)

(2.3.11)

j=1

Here our objective is to minimize the initial inputs in stage-1 and external inputs in the second stage. So as the objective function includes both inputs. The corresponding model is given as:

52

(1)

"

M inimize

etwork ρN jo

(2)

m m −,(1) −,(2) 1 X si 1 X si = t− + (2) m(1) i=1 x(1) m i=1 x(2) ijo ijo

Subject to Constraints; s 1 X s+ r t+ =1 s r=1 yrjo n X j=1 n X j=1 n X j=1 n X j=1

(2.3.12)

(1)

−,(1)

= txijo ;

(2)

−,(2)

= txijo ;

λj xij + si

µj xij + si

(1)

(2)

µj yrj − s+ r = tyrjo ; λj zdj =

n X

#

i = 1, ..., m(1) i = 1, ..., m(2) r = 1, ..., s

µj zdj ; d = 1, ..., D

j=1

λj , µj , s− , s+ ≥ 0; ∀j, i, r and t > 0 Similarly, when compared to the previous models, this model gives the network etwork efficiency of type-III of two stage DEA structure. If ρN = 1, then corresponding j DMU is SBM efficient, otherwise it is inefficient. This process has been supported with numerical illustration of real-life data set. 2.3.3.1

Numerical Illustration on Type-III

This section presents the real life example based on T ypeIII production process. For this type of production process we take example from Li et al. (2012). The example is of regional R&D process of 30 provincial level regions in china. The Figure 2.4 shows the regional R&D process:

Figure 2.4: Illustration on Type-III of two-stage processes 53

The inputs of the first stage are R&D personnel (x1), R & D Expenditure (x2) and proportions of regional science and technology funds in regional total financial expenditure (x3), where as the two outputs of the first stage are Patents (z1) and papers (z2). These two outputs of first stage are also inputs of second stage hence called as intermediate measures. The second stage also has an input of contract value (x(2) ) other than intermediate measures. The final outputs are GDP (Y1), total exports (Y2), urban per capita annual income (Y3) and gross output of hightech industry (Y4). It can be seen from the table that columns two and three represents the stage one and stage two efficiency scores respectively. The system efficiency of the network process is presented in the column four. A DMU which is inefficient in any of the stages can not be network efficient. Table 2.3.3: Results based on type-III process Regions Beijing Chongqing Shanghai Tianjin Anhui Fujian Gansu Guangdong Guizhou Hainan Hebei Heilongjiang Henan Hubei Hunan

ρ1∗ j 1.000 1.000 1.000 0.613 0.614 0.492 1.000 1.000 0.895 1.000 0.929 0.817 0.783 0.879 1.000

ρ2∗ j 0.113 0.253 0.515 0.435 0.369 1.000 0.265 1.000 1.000 1.000 1.000 0.274 1.000 0.372 0.382

ρjN etwork∗ 0.163 0.234 0.545 0.357 0.190 0.498 0.218 1.000 0.819 1.000 0.461 0.234 0.561 0.307 0.302

Regions Jiangsu Jiangxi Jilin Liaoning Qinghai Shandong Shanxi Shanxi Sichuan Yunnan Zhejiang Guangxi Inner Mongolia Ningxia Xinjiang

ρ1∗ j 0.836 0.538 0.602 0.666 0.416 0.644 0.490 1.000 1.000 1.000 0.779 1.000 0.336 0.404 1.000

ρ2∗ j 1.000 1.000 0.596 0.359 1.000 1.000 0.443 0.284 0.667 0.508 1.000 1.000 1.000 1.000 1.000

ρjN etwork∗ 1.000 0.438 0.375 0.282 0.422 0.656 0.177 0.311 0.613 0.389 0.808 1.000 0.336 0.407 1.000

From Table 2.3.3 the company Beijing is efficient in first stage whereas it is inefficient in the second stage, therefore, it can not be network efficient. To become network efficient it has to reduce input and output slacks in second stage only as its efficiency is equal to one in the first stage. Table 2.3.3 shows that the companies Hainan, Zhejiang and Xinjiang are efficient in both stages as well as in system efficiency, while as other DMU other than these are either inefficient in stage first or in second or in both.

54

2.3.4

Type-IV Structure of Two-stage Process

The Type-IV is the last and most general structure of two-stage series processes. In this type, unlike other types of both stages have some external impact on the production process. At the first stage, some outputs leave the system while at second stage some external inputs enter the system. The process can be well understood by the graphical representation in Figure 2.5. This process consists of transforming m(1) initial inputs into two types of outputs (Z

Figure 2.5: Type-IV of series type two-stage DEA processes intermediates and y (1) external outputs), among which one (y (1) ) leaves the system while the other (z) is used as input for second stage along with some other external inputs (x(2) ) at second stage to produce final outputs (y (2) ). Thus, first stage consists of one type of inputs and two types of outputs, whereas, the second stage consists in employing two types of inputs to produce final single type output. The non-radial SBM model for first stage with one input constraint and two output constraint is as follows; (1)

M inimize

(1) ρjo

m −,(1) 1 X si = t − (1) m i=1 x(1) ijo

Subject to Constraints; " # D s(1) +,(1) X 1 X s+ 1 s r d t+ + (1) =1 (1) D d=1 xdjo s r=1 yrj o n X (1) −,(1) (1) λj xij + si = txijo ; i = 1, ..., m(1) j=1 n X

λj zdj − s+ d = tzdjo ;

d = 1, ..., D

j=1

55

(2.3.13)

n X

(1)

(1)

λj yrj − sr+,(1) = tyrjo ; r = 1, ..., s(1)

j=1

λj , s− , s+ ≥ 0; ∀j, i, d, r and t > 0 Here we have two outputs in the first stage which is the reason of extra constraint as similar in first stage of type-II process. Similarly, we can formulate the SBM model for second stage where we have external inputs along with intermediate measure. The SBM model for second stage is given by: (2)

"

(2)

M inimize ρjo

D m −,(2) 1 X s− 1 X si d = t− + (2) D d=1 zdjo m i=1 x(2) ijo

#


(2.3.14)

(2)

s +,(2) 1 X sr =1 t + (2) (2) s r=1 yrj o n X µj zdj + s− d = tzdjo ; j=1 n X j=1 n X

(2)

−,(2)

µj xij + si (2)

d = 1, ..., D (2)

i = 1, ..., m(2)

(2)

r = 1, ..., s(2)

= txijo ;

= tyrjo ; µj yrj − s+,(2) r

j=1 − + µj , s− d , si , sr ≥ 0; ∀j, d, i, r and t > 0

Here our objective is to minimize simultaneously the initial inputs at first stage and external inputs at the second stage So the objective function consists of both the types of inputs. Here, in this process we have final outputs in stage first and external inputs at the second stage. Thus the corresponding model optimizes all the inputs and outputs in the network process. Now to connect stage first and stage second we have to add connecting constraints like in others to formulate the overall efficiency model. For intermediate P P measures we must have nj=1 λj zdj = nj=1 µj zdj ; d = 1, · · · , D. Additionally, to formulate the network SBM model, output of the first stage cannot be neglected, thus, following output constraint has to be added to the network model.

56

n X

(1)

(1)

λj yrj − s+,(1) = tyrjo ; r = 1, ..., s(1) r

(2.3.15)

j=1

Similarly, the external input at the second stage cannot be neglected. In order to consider its impact on the overall network process we have to include the following constraint to the model; n X

(2)

(2)

r = 1, ..., s(2)

= tyrjo ; µj yrj − s+,(2) r

(2.3.16)

j=1

The network model for the type-IV of two-stage production process thus can be formulated as follows; "

etwork M inimize ρN jo

(1)

(2)

m m −,(1) −,(2) 1 X si 1 X si + = t− m(1) i=1 x(1) m(2) i=1 x(2) ijo ijo

Subject to Constraints; " # s(1) +,(1) s(2) +,(2) 1 X sr 1 X sr t + (1) + (2) =1 (1) (2) s r=1 yrj s r=1 yrj o o n X (1) −,(1) (1) λj xij + si = txijo ; i = 1, ..., m(1)

#

(2.3.18)

j=1 n X

−,(2)

(2)

µj xij + si

(2)

= txijo ;

i = 1, ..., m(2)

j=1 n X j=1 n X j=1 n X

(1)

(1)

(2)

(2)

λj yrj − sr+,(1) = tyrjo ; r = 1, ..., s(1) µj yrj − s+,(2) = tyrjo ; r λj zdj =

j=1

n X

r = 1, ..., s(2)

µj zdj ; d = 1, ..., D

j=1 −

+

λj , µj , s , s ≥ 0; ∀j, i, r and t > 0 Similarly to the previous models this model gives the network SBM efficiency of type-IV of two stage DEA structure. If ρ∗jo = 1, then corresponding DMU is SBM

57

efficient; otherwise, it is inefficient.

2.4

Multi-Stage Series Process

The above process are limited only to two-stages. There could be process with more stages. A multi-stage series process is the generalization of type-iv to ’Q’ such stages. The network model or divisional model for any number of stages can be derived from the generalized model by simply substituting ’Q’ with the number of stages. To propose the network model for ’Q’ stages and divisional model for q th stage, one should be familiar with the following notations. • q = 1, ..., Q is the index of q th stage • Xjq = {xqij , i = 1, ..., m}, external input vector of j th DMU in the q th stage. q−1 , d = 1, ..., D}, Intermediate vector of j th DMU in the q th stage. • Zjq−1 = {zdj q • Yjq = {yrj , r = 1, ..., s}, External output vector of j th DMU in q th stage. −,(q)

• si

is the vector of input slacks for j th DMU in q th stage.

−,(q−1)

is the vector of intermediate slacks for j th DMU in q th stage.

+,(q+1)

is output slacks vector for j th DMU in q th stage.

• sd

• sd

+,(q)

• sr

is the vector of output slacks for j th DMU in q th stage.

(q)

• ρj = Divisional efficiency score of j th DMU at q th stage. etwork = The overall network efficiency of Q stages. • ρN j

A multi-stage process is the generalization of series type two-stage processes to series type of multi-stage with Q stages. The multi-stage series structure is displayed in Figure 2.6. The Figure 2.6 represents a series type multi-stage production process with Q stages where every stage except first and last uses two types of inputs to produce two types of outputs. For example q th stage uses two types of inputs, one type from previous q − 1th stage ( z (q−1) ) as intermediate measure and other type from from outside at the same stage x(q) to produce two type of outputs as z q which

58

Figure 2.6: Multi-stage series production process will be further used at q + 1th stage and y (q) external outputs. The generalized SBM model for the q th stage is as follows;  (q)

M inimize ρjo = t −

(q) m X

−,(q) si  1 m(q) i=1 x(q) ijo

+

(q−1) DX

1 D(q−1)

d=1



−,(q−1) sd  (q−1) zdjo

Subject to Constraints;   (q+1) (q) DX s +,(q+1) +,(q) sd 1 X sr  1 =1 + t +  (q+1) (q+1) (q) (q) D s z y r=1 rjo d=1 djo n X j=1 n X j=1 n X j=1 n X j=1 n X j=1

(q) (q)

−,(q)

λj xij + si (q−1) (q−1) zdj

λj

(q)

i = 1, ..., m(q)

= txijo ; −,(q−1)

+ sd

(q) (q)

(q−1)

= tzdjo

(q) (q)

+,(q)

(q−1) (q−1) λj zdj

r = 1, ..., s(q)

(q)

d = 1, ..., D(q)

=

= tzdjo ;

n X

; d = 1, ..., D(q−1)

(q)

λj yrj − sr+,(q) = tyrjo ; λj zdj − sd

(2.4.1)

(q) (q−1)

λj zdj

; d = 1, ..., D(q−1)

j=1

λj , si , sd , sr ≥ 0; ∀j, i, d, r, and t > 0 The last constraint in 2.4.1 other than non-negative conditions is for connecting process. With this constraint we assume that the weighted output from the previous q − 1th stage is equal to the weighted input for the q th stage. For the overall network model we have consider all these links of all stages to formulate the network SBM 59

model. The SBM model for multi-stage production process is given in 2.4.2 etwork M inimize ρN jo

  (q) Q m −,(q) X X si 1  = t− (q) (q) m x q=1 i=1 ijo

Subject to Constraints;   Q s(q) +,(q) X X 1 sr  =1 t+ (q) (q) s r=1 yrj q=1 o n X

(q) (q) λj xij

+

−,(q) si

−

s+,(q) r

(2.4.2)

=

(q) txijo ;

i = 1, ..., m ; q = 1, · · · , Q Q Input Consts.

=

(q) tyrjo ;

r = 1, ..., s ; q = 1, · · · , Q Q Output Consts.

(q)

j=1 n X j=1 n X j=1

(q) (q) λj yrj

(q−1) (q−1) λj zdj

=

n X

(q) (q−1) λj zdj ;

(q)

d = 1, ..., D

(q−1)

; q = 1, · · · , Q Q connecting Consts.

j=1

λj , si , sd , sr ≥ 0; ∀j, i, d, r, and t > 0 The mathematical model 2.4.2 is a generalized SBM network model for the series type DEA processes with Q stages. For any number of stages, the corresponding network model could be derived from the generalized model 2.4.2 by simply substituting Q with the number of stages. For instance, to formulate network model for a production process with five stages, substitute Q by five in the generalized model leads to network model for series type five stage DEA process. This model is solved for all DMUs to get the respective efficiency scores.

2.5

Conclusions

The prime objective of this chapter is to present non-radial slack-based models for all possible types of two-stage series production processes. The structure of all four types of two-stage series-type production processes is given in their respective sections in the chapter. We formulated the SBM models for the individual stages as well as for network processes in all four types production processes with respective real-world problems for type-I and type-III. However, the examples on Type-II and type-IV are not analyzed numerically due to the reason of unavailability of data. The chapter also includes the generalization the two-stage process to multi-stage 60

production process with Q stages and presented the generalized slack-based model for the q th stage as well as network SBM model for the overall multi-stage production process.

61

Chapter 3

SBM Double Frontiers in Two-Stage Production Processes 3.1

Introduction

The basic DEA models such as CCR, BCC or any other non-radial model evaluates the best relative efficiency while assuming the favorable condition for the production processes. All such models provide a best relative efficiency score by maximizing the ratio of virtual set of outputs to the virtual set of inputs. This measure is some times referred as optimistic efficiency score or simply optimistic efficiency. In other words, all the radial and non-radial models tries to maximize the efficiency by assigning most preferable weigh through LPPs to the set of DMUs under consideration. If the optimistic relative efficiency comes out to be one for a particular DMU, it is referred as ’optimistic efficient’ DMU or DEA efficient; otherwise, it is optimistic non-efficient or DEA inefficient. Optimistic efficient DMUs always outperforms the optimistic non-efficient DMUs. The optimistic frontier is obtained by assigning the most favorable weights through linear programming problems, and improves the efficiency of optimistic non-efficient DMUs by enhancing their current output levels or decreasing their current input levels depends upon the orientation of the problem. On the other hand, if one minimizes the same problem of the ratio of virtual set of outputs to the virtual set of inputs, the resulting efficiency is called pes62

simistic efficiency or worst relative efficiency. By minimizing the same problem, one could obtain totally opposite frontier as obtained by optimistic DEA model, because by minimizing the problem, each DMU under evaluation will acquire the most unfavorable weights. Unlike the optimistic case, a DMU is said to be pessimistic in-efficient if its relative efficiency is equal to one. Otherwise, it is called pessimistic non-inefficient or DEA non-inefficient. Pessimistic non-inefficient DMUs always outperforms the pessimistic inefficient DMUs. Thus, optimistic and pessimistic frontiers comprise the two extreme frontiers for each DMU. This chapter is on the proposed optimistic and pessimistic SBM models for two-stage production processes. The two opposite extreme frontiers, optimistic and pessimistic of a DMU can be obtained respectively by maximizing and minimizing the same LPP for the corresponding DMU. The efficiency of a DMU can be evaluated based on any of the two frontiers depends on the conditions of the production process. For instance, if the conditions are favorable to the production process, optimistic frontier can be considered, otherwise, pessimistic frontier. Usual DEA models assume favorable conditions for the production process. If the conditions regarding the favorability and non-favorability of the production process are unknown, then any evaluation method that considers only one of them is considered to be biased. In such cases, the efficiency estimates should be based on both optimistic and pessimistic frontiers. Any approach that determines the efficiency on both frontiers is called as double frontier DEA (Wang and Chin, 2009, 2011). Entani et al. (2002) initially attempted to measure the performance of a DMU from both optimistic and pessimistic perspectives. Their model provides interval efficiency based on DEA and Inverted DEA models. They failed to determine a scalar measure based on two extreme frontiers. Furthermore, their model determines the lower bound of efficiency only based on one input and one output. Several attempts were made to overcome these problems [Wang et al. (2007); Wang et al. (2008); Azizi (2011); Azizi and Wang (2013)]. Paradi et al. (2004) identified the worst performers to represent the inefficient frontier and its application on credit risk evaluation. Jahanshahloo and Afzalinejad (2006) ranked the DMUs relative to full in-efficient frontier. They designed the CCR, BCC and SBM models based on the inefficient frontier to evaluate worst relative efficiency. Shuai and Li (2005) proposed a hybrid approach that predicts the failure of firms through worst practice DEA and rough set approach. Wang and Luo (2006) 63

fused the TOPSIS method with DEA and introduced two virtual DMUs namely ideal DMU and anti-ideal DMU. They presented the models to optimistic and pessimistic efficiency measures based on these virtual DMUs and finally, they rank the DMUs based on a common measure of two distinct efficiencies. Amirteimoori (2007) proposed an efficiency measure using ideal and anti-ideal indices which are formed on the basis of optimistic and pessimistic frontiers respectively. The fundamental reason of these two indices is to maximize the weighted distance function relative to optimistic and pessimistic production frontiers. Wang et al. (2007) obtained optimistic and pessimistic efficiency estimates independently and for overall efficiency measure. They have considered the geometric mean of two distinct and opposite estimates. The overall efficiency integrates both optimistic and pessimistic for each DMU and so is more comprehensive than any one of them taken individually. Similarly, Wang and Chin (2009) proposed a model to rank the advanced manufacturing technologies based on optimistic and pessimistic frontiers simultaneously. Liu and Chen (2009) proposed the worst practice frontier in non-radial SBM form and named it as WPFSBM (Worst practice frontier) to identifying bad performers in the most unfavorable scenarios. Azizi and Ajirlu (2011) proposed a novel pair of DEA models for evaluating efficiency in imprecise interval data. Similarly, Wang and Chin (2011) measured the optimistic and pessimistic efficiency in fuzzy environments and proposed fuzzy expected value approach to measure expected values of inputs and outputs. Azizi et al. (2015) presented the optimistic and pessimistic efficiency in SBM perspectives for evaluation of DMU under consideration with imprecise data. They evaluated SBM efficiency with respect to both efficient and inefficient frontiers. Further, geometric average of the two opposite efficiency scores of same DMU is used to determine the DMU with the best performance. Roozbeh et al. (2015) employed the optimistic and pessimistic models with negative data to obtain Most Productive scale size (MPSS) DMUs. Aldamak et al. (2016) contributed non-convex optimistic and pessimistic models by applying Free Disposal Hull (FDH) technology. The mentioned studies of double frontiers consider the DMUs as Black-boxes. There are numbers of real-life problems resembling with two-stage or multi-stage DEA processes. A Detailed discussion on such type of processes is given in the previous chapter. In this chapter, we present our proposed SBM models for double frontiers in two-stage production processes. This chapter is unfolded as follows; Section 3.2 is devoted to double frontiers 64

wherein the subsections 3.2.1 and 3.2.2 respectively represent the radial and nonradial models for double frontiers. Section 3.3 presented the SBM double frontiers in case of two-stage production processes, where its subsections are devoted separately to the first stage, second stage and network stage. An overall measurement of efficiency based on double frontiers is provided in section 3.4. A numerical illustration based on the proposed models is given in section 3.5. The conclusion of the chapter is presented in the last section 3.6.

3.2

DEA Double Frontiers

The double frontier consists of two opposite and extreme optimistic and pessimistic frontiers. These two extreme frontiers for a DMU for a particular period of time can be obtained by assigning most favorable and unfavorable weights through LPP to different DMUs. The optimistic frontier is obtained by assigning most favorable weights and pessimistic frontier by most unfavorable weights. The graphical representation of these two frontiers with a single input and single output for same data set is given in Figure 3.2.

Figure 3.1: Optimistic and Pessimistic DEA Frontiers The left side of Figure 3.2 represents the optimistic frontier, where DMU are evaluated with respect to best practice frontier. Each inefficient DMU, in this case, tries to move towards the frontier. On the other hand, the right graph of Figure 3.1 represents the pessimistic frontiers, where DMUs are evaluated with respect to 65

worst practice frontier. In this case, each DMU tries to move away from the frontier. If a decision maker knows the nature of production process, he can accordingly use optimistic or pessimistic frontier. In other words, if there are favorable conditions for production process then for efficiency evaluation, optimistic frontier should be used otherwise, pessimistic frontier should be used. If the conditions are unknown, double frontier will provide better results. This sort of system will evaluate the efficiency based on both the frontiers and is graphically represented in Figure 3.2

Figure 3.2: DEA Double Frontiers The models for optimistic and pessimistic frontiers can be formulated in the radial as well as in the non-radial form. Here we first present double frontier in the radial case and then in SBM form in next succeeding sections.

3.2.1

Radial Models for DEA Double Frontiers

Suppose there are n DMUs available to be evaluated each using m inputs to produce s outputs. Let xij , (i = 1, ..., m) and yrj , (r = 1, ..., s) respectively denotes the input and output values of DM Uj , (j = 1, .., n) which are known and nonnegative. Then the efficiency of any DMU is evaluated as follows; Ps ur yrj M aximize θj = Pr=1 m i=1 vi xij

(3.2.1)

Where ur , (r = 1, ..., s) and vi , (i = 1, ..., m) are respective weights of s outputs and m inputs which are to be determined.

66

3.2.1.1

Optimistic Radial Model

The optimistic relative efficiency of a DMU in the set of DMUs can be measured by the following CCR model (Charnes et al., 1978). Ps ur yrjo M aximize θjo = Pr=1 m i=1 vi xijo Subject to Constraints; Ps ur yrj Pr=1 ≤ 1, m i=1 vi xij

(3.2.2) ∀ j = 1, ..., n

ur , vi ≥ 0 ∀ r and i The Fractional Programming Problem (FPP) 3.2.2 maximizes the efficiency of a particular DM Ujo subject to the condition that this ratio for all DMUs should be less than or equal to one. This fractional programming can be translated to a linear one by applying Charnes Cooper transformation (See Appendix-II). M aximize θjo =

s X

µr yrjo

r=1

Subject to Constraints; m X νi xijo = 1 i=1 s X r=1

µr yrj −

m X

(3.2.3)

νi xij ≤ 0,

∀ j = 1, ..., n

i=1

µr , νi ≥ 0 ∀ r and i This model is solved for all n DMUs under study. If θjo = 1 for the evaluated DMUU then it is said to be efficient; otherwise, it is called as inefficient. Optimistic efficiency is same as CCR efficiency.

67

3.2.1.2

Pessimistic Radial Model

The pessimistic model for DMU can be proposed by minimizing fractional programming problem 3.2.1 Wang et al. (2007). Ps ur yrjo M inimize φjo = Pr=1 m i=1 vi xijo Subject to Constraints; Ps ur yrj Pr=1 ≥ 1, m i=1 vi xij

(3.2.4) ∀ j = 1, ..., n

ur , vi ≥ 0 ∀ r and i This model differs from the basic CCR model in the sense that it minimizes the efficiency of a particular DMU relative to others within the range of no less than one, whereas CCR model maximizes the efficiency with the range zero to one. This FPP model can also be transformed into linear one like above model by Charnes Cooper transformation as follows; M inimize φjo =

s X

µr yrjo

r=1


(3.2.5)

i=1 s X r=1

µr yrj −

m X

νi xij ≥ 0,

∀ j = 1, ..., n

i=1

µr , νi ≥ 0 ∀ r and i This model is solved for all DMUs under study and optimal value is always greater than one. The DMUs whose φ∗ = 1 constitutes the inefficient frontier and are called as pessimistic inefficient units. whereas, If the optimal value for any DMU comes out to be greater than one, (φ∗ > 1) then DMU is said to be the pessimistic non-inefficient DMU.

68

3.2.2

Non-Radial SBM Models for DEA Double Frontiers

The SBM model of optimistic and pessimistic frontiers for the DMU jo , (jo ∈ J) with same input and output vectors as defined in the section 3.2.1 are presented respectively in the following subsections; 3.2.2.1

Optimistic Non-Radial SBM Model

The SBM model as given by Tone (2001) evaluates the relative efficiency while assigning the most favorable weights to the DMUs in non-radial form, and hence forms the optimistic frontier for evaluation method. The Production Possibility Set (PPS) P can be defined as follows; P = {(x, y)/x ≥

n X j=1

Λj xij , y ≤

n X

Λj yrj , Λj ≥ 0}

(3.2.6)

j=1

Where Λ ∈ 0.

The model 3.2.9 is a LPP and can be solved easily for all of the DMUs. After evaluating the optimal values of this model, the values of original model 3.2.8 can

70

be obtained through following substitutions. τj∗0 = ρ∗j0 ; Λ∗ =

λ∗ s−∗ s+∗ −∗ +∗ , S = and S = t∗ t∗ t∗

(3.2.10)

Similarly, with the radial model, this model has to solve for all n DMUs to get non-radial efficiency score. All the DMUs whose optimal score comes out to be one constitute the optimistic or best practice frontier and DMUs are to be evaluated with respect to it. The important property of is that it is independent of the unit of measurement of inputs and outputs. Furthermore, the SBM score is monotonically decreasing for every positive slack. Therefore, the larger the value of the more is the performance of a DMU. 3.2.2.2

Pessimistic Non-Radial SBM Model

The PPS (Pb) with CRS assumption for evaluating the pessimistic relative efficiency for a set of DMUs is completely opposite to the optimistic one and is given as follows; Pb = {(x, y)/x ≥

n X

Λj xij , y ≤

j=1

n X

Λj yrj , Λj ≥ 0}

(3.2.11)

j=1

The inefficient PPS (Pb) is a closed and convex set and extreme points constitutes inefficient frontier. Each DMU in the set is evaluated with respect to the inefficient frontier. Based on the inefficient PPS following SBM model has been proposed for measuring the pessimistic efficiency.

M aximize φj0 =

1+

1 m

1−

1 s

Si− i=1 xij0

Pm

Sr+ r=1 yrj0

Ps

Subject to Constraints; n X πj xij + Si− = xij0 ; i = 1 , 2 , 3 , . . . m.

(3.2.12)

j=1 n X

πj yrj − Sr+ = yrj0 ; r = 1 , 2 , 3 , . . . s.

j=1

πj ≥ 0; Si− ≥ 0; Sr+ ≥ 0, ∀j, i, r. The model assigns a different weight π compared to optimistic model for each DMU 71

under study, because of the fact that pessimistic model deals with unfavorable conditions. The numerator and denominator of the objective function are always greater than one and less than one respectively, hence makes φ∗ ≥ 1 for each DMU. This measure is monotonically increasing function for all positive slacks. The FPP can be transformed into following LPP; m

1 X s− i M aximize ρj0 = t + m i=1 xij0 Subject to Constraints; s 1 X s+ r t− =1 s r=1 yrj0 n X j=1 n X

λj xij + s− i = txij0 ; i = 1 , 2 , 3 , . . . m.

(3.2.13)

λj yrj − s+ r = tyrj0 ; r = 1 , 2 , 3 , . . . s.

j=1 + λj ≥ 0; s− i ≥ 0; sr ≥ 0, ∀j, i, r; and t > 0.

Where the optimal solutions for the original FPP can be obtained through the substitution as were done in optimistic case. If ρj0 = 1 then the correspond DMU is pessimistic inefficient; otherwise, it is said to be pessimistic non-inefficient (if ρj0 > 1). Obliviously, a pessimistic non-inefficient DMU is not necessarily optimistic efficient.

3.3

SBM Double Frontiers for Two-Stage DEA Processes

As discussed in chapter two, DMUs can have a two-stage structure where the inputs are transformed into outputs via the intermediates as shown in Figure 3.3. For example, banks use labor and assets to generate deposits which are in turn used to generate loan income and loans recovered (Chen and Zhu, 2004). In this elementary case of the two-stage production process, each DMU transforms m external inputs denoted by xij , (i = 1, ..., m) to final s outputs denoted by yrj , (r = 1, ..., s) via D intermediate measures as zdj , (d = 1, ..., D). A DMU is said to be

72

Figure 3.3: Two-Stage DEA Structure overall efficient if and only if it is efficient in both the stages. The optimistic and pessimistic frontiers in non-radial SBM form for first stage, second stage and network processes are presented in the following subsections.

3.3.1

Double Frontiers for the First Stage

In this section, we present the SBM models for evaluating optimistic and pessimistic frontiers for the first stage of the network process while ignoring the second stage. The first stage of the process utilizing m inputs to produce D intermediate measures. Thus, based on these inputs and intermediates, Optimistic and pessimistic frontiers in SBM form can be obtained as follows; 3.3.1.1

Optimistic Non-Radial SBM Model for First Stage

The best relative efficiency or optimistic efficiency is always same as conventional DEA models regardless of any of the approach. Thus, the SBM model for the first stage to evaluate optimistic efficiency will be same as usual SBM but with different output constraints. The Optimistic SBM model in linear form for a DM Ujo to evaluate its optimistic efficiency at the first stage with m inputs denoted by xij , (i = 1, ..., m) and D outputs(intermediates)denoted by zdj , (d = 1, ..., D) is as follows;

73

m

M inimize

(1) ρ j0

1 X s− i = t− m i=1 xij0

Subject to Constraints; D

t+ n X

1 X s+ d =1 D d=1 zdj0 λj xij + s− i = txij0 ; i = 1 , 2 , 3 , . . . m.

(3.3.1)

j=1 n X

λj zdj − s+ d = tzdj0 ; d = 1 , 2 , 3 , . . . D.

j=1 + λj ≥ 0; s− i ≥ 0; sd ≥ 0, ∀j, i, d; and t > 0. (1)

Where the superscript in ρj0 represents the stage of the process. The numerator of the objective function is always less than or equal to one and the denominator as greater than or equal to one , hence the the overall value is always less than one. The set of DMUs whose optimal value comes out as one represents the optimistic frontier and all the other DMUs are evaluated with respect to the this optimistic frontier. 3.3.1.2

Pessimistic Non-Radial SBM Model for First Stage

The SBM model for evaluating the pessimistic efficiency or worst relative efficiency for a set of DMUs at the first stage with same inputs and outputs quantities as in optimistic case, could be formulated as follows;

74

m

M aximize

(1) ρj0

1 X s− i = t+ m i=1 xij0

Subject to Constraints; D

t− n X

1 X s+ d =1 D d=1 zdj0 λj xij + s− i = txij0 ; i = 1 , 2 , 3 , . . . m.

(3.3.2)

j=1 n X

λj zdj − s+ d = tzdj0 ; d = 1 , 2 , 3 , . . . D.

j=1 + λj ≥ 0; s− i ≥ 0; sd ≥ 0, ∀j, i, d; and t > 0.

The optimal objective function value of this model is no less than one. If the value for any evaluated DMU is one, it is said to be pessimistic inefficient DMU and the set of such DMUs comprises the inefficient frontier. The inefficient frontier like optimistic one covers all DMUs under it but in opposite direction and all the DMUs are evaluated with respect to it. If the optimal value occurs greater than one for a particular DMU, then the DMU is said to be pessimistic non-inefficient. Thus, models 3.3.1 and 3.3.2 represent the two extreme frontiers for the first stage of the two-stage production process.

3.3.2

Double Frontiers for the Second Stage

In the second stage of two-stage production process, the outputs of first stage are used as inputs to produce the final outputs that leave the system. Thus, the second stage uses D (zdj , d = 1, ..., D ) inputs(intermediates) to produce s (yrj , r = 1, ..., s ) final outputs. The optimistic and pessimistic frontiers for the second stage are given in following subsections. 3.3.2.1

Optimistic Non-Radial SBM Model for Second Stage

The SBM model for evaluating best practice efficiency or optimistic efficiency through non-radial SBM approach for the second stage with D inputs and s outputs can be formulated as follows;

75

D

M inimize

(2) ρ j0

1 X s− d = t− D d=1 zdj0

Subject to Constraints; s 1 X s+ r =1 t+ s r=1 yrj0 n X j=1 n X

µj zdj + s− d = tzdj0 ; d = 1 , 2 , 3 , . . . D.

(3.3.3)

µj yrj − s+ r = tyrj0 ; r = 1 , 2 , 3 , . . . s.

j=1 + µj ≥ 0; s− d ≥ 0; sr ≥ 0, ∀j, d, r; and t > 0.

The model 3.3.3 is the SBM pessimistic DEA model for the second stage. The weights of the second stage may not be same as in the first stage hence different weights should be assigned for DMUs in the second stage. The variable µj is the weight of DMUs in the second stage. Here intermediates will also be assigned by different weights as the stage is considered to be as independent of its previous stage. 3.3.2.2

Pessimistic Non-Radial SBM Model for Second Stage

As mentioned above, the second stage transforms the intermediates into final outputs. To obtain its pessimistic efficiency frontier through non-radial approach following SBM model can be used.

76

D

M aximize

(2) φ j0

1 X s− d = t+ D d=1 zdj0

Subject to Constraints; s 1 X s+ r =1 t− s r=1 yrj0 n X j=1 n X

µj zdj + s− d = tzdj0 ; d = 1 , 2 , 3 , . . . D.

(3.3.4)

µj yrj − s+ r = tyrj0 ; r = 1 , 2 , 3 , . . . s.

j=1 + µj ≥ 0; s− d ≥ 0; sr ≥ 0, ∀j, d, r; and t > 0.

The pessimistic model belongs to the second stage and as such different weights will be assigned to DMUs to obtain the pessimistic frontier. The models 3.3.3 and 3.3.4 represent independently the optimistic and pessimistic (double frontiers) for the second stage. It is possible to evaluate the efficiency scores based on double frontiers for the divisional stages. However, the divisional score refers to a particular stage not the whole network system. The system efficiency based on double frontiers can be evaluated by formulating the optimistic and pessimistic SBM models and then by taking their geometric average. Lets first introduce the networks models in optimistic and pessimistic case.

3.3.3

DEA Double Frontiers for the Network System

The network system represents the whole system of the process. A DMU efficient in both the stages will be network efficient, whereas any DMU which is not efficient in all stages will not be network efficient. Based on the network model a DMU can be ranked as it evaluated the overall efficiency of the process. In a network model, inputs are to be transformed into outputs via some intermediaries. Neglecting the intermediaries in efficiency evaluation will clearly be erroneous. To include these measures into efficiency evaluation, network DEA models have been formulated. The importance of network efficiency models is not only to include the intermediate effects in the efficiency evaluation but also enables to locate the source of inefficiency. In order to include the intermediaries into the process, the relationship between the 77

stages should be identified. This relationship between two stages can be found in the sense that outputs of the first stage are used as inputs for the second stage. In other words, the weighted output of the first stage and weighted input of the second stage must be equal. Thus, in order to formulate the network SBM model, this equality constraint must be added to the constitution. Mathematically, the following constraint must be added to guarantee the connectivity of two stages; n X

λj zdj =

n X

j=1

µj zdj , ∀d = 1, 2, ..., D

(3.3.5)

j=1

The additional constraint 3.3.5 will be included in the set of constraints to formulate network models for both optimistic and pessimistic frontiers. The network models for optimistic and pessimistic frontiers are given in the following subsections. 3.3.3.1

Optimistic Non-Radial SBM Model for Network Process

The network model considers the initial inputs and final outputs of the process while keeping the connectivity assumption between the stages. The network model uses m initial inputs to produce s final outputs. The optimistic model for a DM Ujo will be formulated as follows; m

M inimize

etwork ρN j0

1 X s− i = t− m i=1 xij0

Subject to Constraints; s 1 X s+ r t+ =1 s r=1 yrj0 n X j=1 n X

λj xdj + s− i = txij0 ; i = 1 , 2 , 3 , . . . m. µj yrj − s+ r = tyrj0 ; r = 1 , 2 , 3 , . . . s.

j=1 n X j=1

λj zdj =

n X

µj zdj , d = 1 , 2 , 3 , . . . D.

j=1

+ λj ≥ 0; µj ≥ 0; s− i ≥ 0; sr ≥ 0, ∀j, i, r; and t > 0.

78

(3.3.6)

In the optimistic network model 3.3.6, the input and output constraints are assigned by different weight restrictions. For the input constraints, λ’s have been assigned as these belong to the first stage, whereas, for output constraints,µ’s have been assigned as these belong to the second stage. For the connectivity assumption, the weights of intermediaries at two different stages must be equal. The model can be solved for all n DMUs independently to obtain the network production frontier for measuring the optimistic or best practice efficiency of the set of DMUs. 3.3.3.2

Pessimistic Non-Radial SBM Model for Network Process

Like divisional pessimistic frontiers, network pessimistic frontier evaluates the DMUs with respect to the inefficient frontier. The variables for this model except for the weights of DMUs will be same as for optimistic network problem. The difference between the network optimistic and pessimistic models is that the objective function is minimized in former and maximized in the later. The model for pessimistic frontier is as follows; m

M axiimize

etwork φN j0

1 X s− i = t+ m i=1 xij0

Subject to Constraints; s 1 X s+ r t− =1 s r=1 yrj0 n X j=1 n X

λj xdj + s− i = txij0 ; i = 1 , 2 , 3 , . . . m.

(3.3.7)

µj yrj − s+ r = tyrj0 ; r = 1 , 2 , 3 , . . . s.

j=1 n X j=1

λj zdj =

n X

µj zdj , d = 1 , 2 , 3 , . . . D.

j=1

+ λj ≥ 0; µj ≥ 0; s− i ≥ 0; sr ≥ 0, ∀j, i, r; and t > 0.

The model 3.3.7 considers the initial inputs and final outputs of the process hence there is one constraint for each of them. As mentioned above, the input and output constraints are assigned by different weights depends upon whether they belong to first stage or second stage. This model will evaluate worst relative efficiency under most unfavorable conditions for the production process for a set of DMUs 79

etwork = 1, then the DMU will be under evaluation. In the evaluation process, if φ∗N j pessimistic inefficient; otherwise, it is referred as pessimistic non-inefficient.

3.4

Overall Efficiency Based on Double Frontiers

The optimistic and pessimistic frontiers provide two different measures from different points of view, results in different efficiency measurements for the set of DMUs. Both the measures can be used independently if the nature of the process well suited to anyone of them. However, if the nature of the production process is anonymous, using any one of them may lead to erroneous results. In such cases, a measure based on both the frontiers must be utilized. Various attempts have been done to evaluate the overall performance based on both the frontiers. Wang et al. (2007) proposed the geometric average method for evaluation of overall efficiency based on double frontiers. Their study justifies that geometric average is better than the arithmetic average. Wang and Chin (2009) proposed a new approach based on both the optimistic and pessimistic frontiers which consider not only the magnitude of two efficiencies but also their directions. In this study, we use the new approach as proposed by Wang and Chin (2009) for scoring DMUs based on double frontiers. The proposed approach to evaluate the overall performance is as follows; etwork ξj∗N o

etwork ρ∗N jo

= qP n

i=1

etwork ρ∗N i

etwork φ∗N jo

2 + qPn

i=1

etwork φ∗N i

2 ; j = 1, 2, ..., n

(3.4.1)

Where and are the network optimistic and pessimistic efficiency estimates for DMU respectively. The expression 3.4.1 will enable to evaluate the overall efficiency based on double frontiers. The expression will provide a scalar measure of overall efficiency which lies between zero and one, which leads to rank the DMUs according to their respective scores.

3.5

Numerical Illustration.

In this section, we support our models by applying them to real-life examples. The proposed models were applied to 24 non-life insurance companies in Taiwan as studied by Kao and Hwang (2008). They considered the production process of insurance companies as the two-stage process, where the first stage is for premium 80

acquisition and the second stage for profit generation. For premium acquisition in the first stage, the insurance companies have to endure two types of expenses as operational expenses and insurance expenses. The outcomes from the first stage are direct written premiums and reinsurance premiums. These two measures are termed as intermediates as they are further used in the second stage as inputs to produce final outputs. Here, the final outputs of the second stage are underwriting profit and investment profit. Thus, the production process utilizes two inputs to produce two outputs via two intermediates. The complete data-set of non-life insurance companies in Taiwan is given in the appendix. The calculation process of efficiency scores in divisions was done in Excel by LPP with the help of Excel-solver. However, the network efficiency scores were evaluated in MATLAB. The overall efficiency and ranking were done by manual calculations. The results are presented in Table 3.5.1. Table 3.5.1 represents the efficiency estimates of Taiwanese non-life insurance companies. The optimistic and pessimistic estimates were evaluated for the first stage, second stage as well as for the network process. The overall performance was calculated based on the double frontiers of the network process. The final ranking was accordingly done on the basis of overall performance measurement ξj∗N etwork . The results reveal that there were in total five companies which are optimistic efficient in the first stage and four in the second stage. Similarly, there were seven pessimistic inefficient companies in the first stage and four in the second stage. The company Kuo Hua is pessimistic inefficient in both the divisional stages and hence in the network system. An overall efficiency measure based on optimistic as well as pessimistic as given by Wang and Chin (2009) is obtained in column eight under the heading of . These efficiency scores are based on not only the double frontiers but also on the basis of two-stage process. The range of this score is between zero and one, thus enables to rank the companies accordingly. The ranking based on the overall efficiency score was done in the last column of the table. The insurance company Asia got the first rank based on the overall efficiency score followed by Cathay Century, Fubon, Union and so on. On the other hand Mitsui Sumitomo was ranked last based on the overall efficiency scores. The results in the table reveal that whatever the conditions of production are, the company Asia tops the list. In other words, whether there are favorable conditions or unfavorable conditions for the production process, the insurance company Asia remains at top of the list. in contrast, However, being 81

Table 3.5.1: Efficiency Scores and Ranking Based on Double Frontiers DMUs Taiwan Fire Chung Kuo Tai Ping China Mariners Fubon Zurich Taian Ming Tai Central The First Kuo Hua Union Shing kong South China Cathay Century Allianz president Newa AIU North America Federal Royal Sunalliance Asia AXA Mitsui Sumitomo

ρ∗j

Stage-1 φ∗j

0.617 0.605 0.415 0.345 0.313 0.887 0.253 0.320 0.414 0.240 0.455 1.000 0.344 0.261 1.000 1.000 0.185 0.604 1.000 0.596 0.523 0.415 0.624 1.000

1.986 1.938 1.191 1.000 1.313 1.349 1.198 1.118 1.594 1.000 1.000 1.507 1.384 1.204 1.755 4.265 1.000 1.554 2.221 1.000 1.000 1.000 1.627 1.564

ρ∗j

Stage-2 φ∗j

0.344 0.211 1.000 0.175 1.000 0.363 0.500 0.450 0.289 0.650 0.036 0.416 0.260 0.507 0.654 0.380 1.000 0.201 0.306 0.709 0.265 1.000 0.113 0.160

4.374 2.747 4.874 2.111 15.483 4.369 8.182 7.838 1.000 10.054 1.000 5.211 5.862 7.334 4.239 4.239 21.390 2.105 2.751 6.146 2.878 9.216 1.000 1.000

Network ρ∗j φ∗j 0.685 0.612 0.676 0.298 0.752 0.382 0.271 0.270 0.219 0.457 0.161 0.744 0.202 0.280 0.595 0.311 0.349 0.251 0.391 0.536 0.197 0.578 0.412 0.132

3.340 2.763 3.437 1.189 4.264 2.902 2.536 2.776 1.617 2.811 1.000 3.992 2.695 2.486 6.389 6.332 4.040 2.231 5.165 5.356 1.169 8.317 1.000 1.054

ξj∗N etwork

Rank

0.492 0.427 0.493 0.200 0.572 0.330 0.260 0.272 0.187 0.359 0.127 0.554 0.237 0.261 0.615 0.483 0.377 0.234 0.456 0.532 0.153 0.711 0.241 0.117

7 10 6 20 3 13 16 14 21 12 23 4 18 15 2 8 11 19 9 5 22 1 17 24

on the top of the list, the company has the scope to further increase their profits and decrease their costs. The companies at the bottom of the list based on overall efficiency score such as Mitsui Sumitomo, have a greater chance to be vanished off if there prevail the unfavorable conditions for the production process.

3.6

Conclusion

This chapter has attempted to shed light on two-stage processes and double frontiers simultaneously. Such processes are important in many real-life situations, the former enables to locate the source of inefficiency inside the production process and the later determines the nature of production process. In this chapter, we formulated optimistic and pessimistic models in non-radial SBM form for the two82

stage production processes. The overall efficiency score based on double frontiers for network process was evaluated using expression proposed by Wang and Chin (2009) rather than the geometric average of two. We applied our model to the published data-set of Taiwans non-life insurance companies studied by Kao and Hwang (2008). We estimated optimistic and pessimistic efficiency scores for first stage second stage and for network process of all companies. Furthermore, we evaluated an overall efficiency score based on both these extreme and opposite efficiency scores which enable us to rank the companies according to their performance.

83

Chapter 4

Multi-Period Efficiency Evaluation Through Window Analysis & Average SBM 4.1

Introduction

The Conventional DEA models including networking DEA models and double frontier DEA models will evaluate the efficiency of a DMU for a particular time period. Since, an efficient DMU at a particular time period may not be efficient at any other period of time. Thus, it seems interesting to calculate the efficiency change over the different periods. The conventional DEA models, evaluates efficiency for a particular period but fails in assessment of efficiency change in cross sectional and time varying data. In order to handle such cases, an extended DEA model has been proposed named as DEA Window Analysis. Assuming a constant technology, DEA Window Analysis evaluates the efficiency change over the period of time. This chapter aims in presenting the window analysis approach in non-radial SBM form. The approach is supported with the numerical illustration of top ten Indian Cement Companies. We also evaluated the panel data through average SBM approach, which consider the averages of inputs and outputs throughout the span of study period. The estimate of such efficiency scores provides a rough idea about how the DMU have performed during the period of time. Apart from its biased measure of 84

efficiency, it can be used to evaluate the crude measure of efficiency over the period of time. Charnes et al. (1984) developed DEA Window Analysis approach model to monitor the productivity change in cross sectional and time varying data. This approach is based on the rationale of moving averages. The main aim of this approach is to monitor the fluctuations in efficiency scores over time. This approach evaluates the performance of DMUs over time by treating them as dissimilar over different periods. In doing so, the performance of a DMU at any time period could be counterpointed with its own performance at different time periods as well as to the performances of other DMUs in the study set. Thus, the number of data points will increase accordingly and will be more useful in small sample cases. This number is inversely proportional to the window width, i.e., lesser the window width more the data points and greater the window width less the number of points. The window width is referred with the number of time periods included in the analysis in a particular window. The range of window width is somewhere between one and total number of in the study horizon. The range of window width remains constant all over the windows. Though DEA models for evaluating efficiency on a particular period have been enormously applied over the past three decades, DEA Window Analysis comes out relatively rarely in the literature. After the initial study of Charnes et al. (1984), there were several studies using DEA window analysis approach. For example, Carbone (2000) exemplified the use of DEA window analysis to observe the performance trends of semiconductor manufacturing over the period of time. Sueyoshi and Aoki (2001) evaluated the performance of Japanese postal services by combining DEA window analysis with the Malmquist index over the period of fifteen years from 1983 to 1997. Ross and Droge (2002) used DEA window analysis approach to evaluate and detect performance trends over four years of distribution centers. Asmild et al. (2004) used Window Analysis to deal with less number of inputs for efficiency evaluation of Canadian Banking Industry for the period of 1981-2000. Cullinane et al. (2004) employed DEA window analysis to a sample of the world’s major container ports in order to assess their efficiency fluctuations. Sufian (2007) determined the efficiency trends of Singapore commercial banks through window analysis over the period 1993 to 2003. They evaluated the pure technical efficiency changes and scale efficiency change over the period separately. Chung et al. (2008) obtained the optimal quantity of semiconductor fabricators through DEA approach to gain max85

imum efficiency. Further, to ensure long term effectiveness and profit gaining, they employed DEA window analysis to seek the most recommended set of products for manufacturing by measuring the performance over the period of time. They compared the performance of a mix not only with the performance of other mixes but also with its own performance in other periods. Yang and Chang (2009) evaluated the efficiency of only three telecommunication firms through DEA window analysis over the period 2001-2005. Pulina et al. (2010) inquired the relationship between size and efficiency of Italian hospitality sector by evaluating technical and scale efficiency scores via DEA window approach. Gu and Yue (2011) employed DEA window analysis for examining the performance of Chinese listed banks. They examined the performances of banks based on seasonal data over the period 2008 to 2010. Pjevˇcević et al. (2012) examined the performance trends of Serbian sea ports through DEA window analysis with window width as four years over the period 2001-08. Sueyoshi et al. (2013) evaluated the environmental performance with desirable and undesirable outputs through window analysis approach U.S. coal-fired power plants during 1995 to 2007. Wang et al. (2013) studied Chinas regional total-factor energy and environmental efficiency of 29 administrative regions through DEA window analysis over the peˇ riod 2000 to 2008. Repkov´ a (2014) employed DEA window analysis to a sample of commercial banks of Czech Republic to examine the productivity trends during the period 2003 to 2012. Meng et al. (2014) examined the inefficiency of mix energy consumption of sixteen APEC countries through their proposed Rank-adjusted model combined with window analysis. Gamassa and Chen (2017) measure the efficiency of East and West African major ports over time by using the DEA Window Analysis. In the similar way Xu and Chi (2017) examined efficiency stability and trend of sample of U.S. hotels through a dynamic perspective from year 2007 to 2014. This chapter is based on employing the DEA window analysis approach to assess the efficiency fluctuations of top ten Indian cement companies during 2007 to 2016. This also includes Average SBM approach to evaluate the average efficiency scores of DMUs over the study span. The contents of this chapter have the following composition. The section 4.2 presents the Indian cement industry in the context of DEA. The SBM methodology for DEA window analysis is presented in the section 4.3. A brief methodology of Average SBM measure is given in the section 4.4. A numerical illustration of top ten Indian cement companies to observe the efficiency trend over the period 2007 to 86

2016 is presented in section 4.5, where the subsections 4.5.1 and 4.5.2 respectively describes the results based on SBM window analysis and average SBM model. The last section 4.6 is devoted to final conclusion of the chapter.

4.2

DEA in Indian Cement Industry

In recent times, cement has become second most consumed substance after water. It is produced from limestone, shell and clay and processed on the temperature of more than 1000 degree Celsius. It is the main ingredient for construction purposes. Without it concrete construction is impossible. So, the utmost care should be taken to improve the performances of this industry. On the other hand, these industries release large amounts of undesirable (bad) accompanied with desirable (good) outputs such as CO2 , dust, water pollution etc. The amount of these undesirable outputs is very huge in developing countries such as India and China. So, it is important to enhance the production in a sustainable manner, which not only reduce the undesirable outputs but also increase its share in GDP of the country. To achieve high growth rates of GDP in a sustainable manner, India has to place much considerations to enhance its manufacturing sector. The target of the Indian devisers is to attain hastened growth in manufacturing sector in such a way that there is not only increase of industry’s share in GDP but also increase in the world export share. Although Indian cement industry dates back to 1914, the country was mainly depend on imports. However, Indian accompanies grows exponentially in the later half of the twentieth century to reach to worlds second position in terms of installed capacity after china. Figure 4.1 shows the cement production of India in 2013 and its estimated value in 2020.

Figure 4.1: Cement Production of India

87

The pace of growth can be observed from the fact that since 1992 it has quadrupled from around 50Mt/yr to 220Mt/yr which not only fulfills the domestic demand but a amount is exported to several countries across the globe. The industry not only grows in terms of quantity but also in terms of quality as presently it competes with the global competition in lowest energy consumption and low CO2 emissions. Presently, there are 137 and 367 large and small plants respectively as per of census 2011. In this study, we have selected top ten leading producers in order to measure their energy intensity, which is the quantity of energy used per unit of output. Much literature was reported on examining the energy intensity of cement industries across the world. In the Indian context, Bhattacharya and Paul (2001) used a complete decomposition technique to decompose the sectoral changes in energy consumption and energy intensity in India during 1980 to 1996. Mandal (2010) estimated the environmental efficiency with CO2 as undesirable output for interstate data. Their results reveal that Indian cement industry has a potential to expand desirable outputs and contract undesirable outputs with the given inputs. Mandal and Madheswaran (2011) used DEA approach to measure the energy efficiency of Indian cement industry and estimates the factors explaining inter-firm variations between the periods 1989-90 to 2006-07. There is a rich body of research to examine energy intensity across various sectors. There are several approaches to examine the efficiency of a production process, but the way DEA finds the efficiency, it has gained popularity in energy efficiency analysis. This method is a non-parametric and non-stochastic approach for assessment of efficiency for a particular time period. In case of cross sectional and time varying data to evaluate the productivity change over the period, its extended approach called as DEA window analysis can be employed.

4.3

Methodology DEA Window Analysis

In order to capture the variations of efficiency over time, Charnes et al. (1984) has proposed a technique called Window Analysis in DEA. The window analysis will assess the performance of a DMU over time by treating each DMU as a different entity in each time-period. This method allows tracking the performance of a unit over time and providing a better degree of freedom. If a DMU is found to be efficient in one year despite the window in which it is placed, it is likely to be considered

88

strongly efficient compared to its peers. Suppose we have n DMUs (j = 1, · · · , n) under study and each of them transforms m (i = 1, · · · , m) inputs to s (r = 1, · · · , s) outputs over the T, (d = 1, · · · , T ) time Periods. Let DM Ujd represents j th DMU at time d whose m dimensional input vector and s dimensional output vector as follows;  x1d j  2d   xj  d  xj =   ..  ;  .  

xmd j

  yj1d  2d  y  d j  yj  .   .. 

(4.3.1)

yjsd

If a widow starts at time period k(1 ≤ k ≤ T ) with a window width w (1 ≤ w ≤ T − k) Then matrices of inputs and outputs are denoted as follows; 

Xkw

xk1  k+1  x1 =  ..  .

xk2 . . . xk+1 ... 2 .. ... .

xk+w xk+w ... 1 2

   ynk xkn y2k . . . y1k    k+1  y1 y2k+1 . . . ynk+1  xk+1 n   .. ..  ..  ...  ; Ykw =  ..  (4.3.2) . .  .   . y1k+w y2k+w . . . ynk+w xk+w n

Substituting these inputs and outputs into the basic DEA Models will produce the results of DEA window analysis. The input oriented DEA window analysis model for DM Udt under a constant returns to scale (CRS) assumption, is as follows; M inimize θ Subject to Constraints;

(4.3.3)

0

θXd − λ Xkw ≥ 0 0

λ Ykw − Yd ≥ 0 λj ≥ 0; ∀ j = 1, · · · n × w The BCC model formulation for window analysis can be formulated by introducing 0 the convexity constraint Iλ in model 4.3.4 to obtain the measure pure technical efficiency.

89

M inimize θ Subject to Constraints;

(4.3.4)

0

θXd − λ Xkw ≥ 0 0

λ Ykw − Yd ≥ 0 0

Iλ = 1 λj ≥ 0; ∀ j = 1, · · · n × w Here, I is the identity w × n matrix. This model is used to differentiate technical efficiency into pure technical efficiency and scale efficiency. Both the CCR or BCC models provides a radial measure of efficiency. in other words, inefficient DMUs have to optimize their inputs and outputs by a similar proportion to reach the frontier. However, it is not possible in some real-life situations. Slack-based measure/model (SBM) is an approach that optimizes through non-radial approach. SBM models allows DMUs to optimize different inputs and outputs by different proportions. The SBM model for window analysis approach of DM Udt can be formulated as follows; 1 1 − ∗ [ × I]Sd 1− m Xd M inimize τ = 1 1 + × I]Sd 1− ∗ [ s Yd Subject to Constraints;

(4.3.5)

0

Λ Xkw + S− kw = Xd 0

Λ Ykw − S+ kw = Yd Λj ≥ 0; ∀ j = 1, · · · n × w The model 4.3.5 is slack-based model for k th window but in fractional form. This fractional problem can be transformed into linear programming problem by applying

90

Charnes cooper transformation. The LPP in SBM approach is as follows; 1 1 − ∗ [ × I]sd M inimize ρ = t − m Xd Subject to Constraints; 1 1 + t− ∗ [ × I]sd = 1 s Yd

(4.3.6)

0

λ Xkw + s− kw = tXd 0

λ Ykw − s+ kw = tYd λj ≥ 0; ∀ j = 1, · · · n × w and t is non − negative vector Model 4.3.6 is applied for every DMU in all windows to estimate the technical efficiency scores in non-radial measure. The analysis is done from one window to another and the windows are made on the principle of moving averages. i.e. one DMU is coming and other DMU leaves the system. The procedure of making windows is elaborated in the illustration section.

4.4

The Approach of Average SBM

The other way to deal with cross sectional and time varying data is to employ the average DEA models to get the average performance score. Although, this method exhibits several limitations, it evaluates the rough average efficiency score. Here we present the average SBM model to access the average SBM efficiency score. Suppose we have n DMUs to be examined in T time periods, each one using m inputs to produce s outputs. Let ith , (i = 1, · · · , m) input and rth , (r = 1, · · · , s) output of j th (j=1,,n) DMU represents their averages and are denoted by x¯ij x and y¯rj x respectively. The production possibility set P is defined as follows; P = {(¯ x, y¯)/¯ x≥

n X j=1

Λj x¯ij , y¯ ≤

n X

Λj y¯rj , Λj ≥ 0}

(4.4.1)

j=1

The variable Λ is an unknown vector of non-negative weights. We can impose P restriction on Λ as nj=1 Λj = 1 to include variable returns to scale. The PPS 4.4.1 unlike usual PPS considers the averages of all inputs and outputs rather than their actual values during the time of study. The PPS implies y¯ can be produced by x¯ over

91

the duration of time if the given conditions are satisfied. We consider an expression for a particular DMU (xjo , yjo ) from the reference set as; x¯jo = y¯jo =

n X j=1 n X

Λj x¯ij + S¯i−

(4.4.2)

Λj ry ¯ j − S¯r+

j=1

With the condition Λj ≥ 0; S¯i− ≥ 0; S¯r+ ≥ 0. The variables S¯i− and S¯r+ represent the average input excess and average output shortfalls respectively over the period of study time and are commonly referred as slacks. Any DMU can be expressed by the equation 4.4.2. The mathematical model in fractional form to evaluate average SBM score is given as; 1− M inimize τj0 =

1+

S¯i− i=1 x ¯ij0 Ps S¯r+ 1 r=1 y¯rj0 s

1 m

Pm

Subject to Constraints; n X Λj x¯ij + S¯i− = x¯ij0 ; i = 1 , 2 , 3 , . . . m. j=1 n X

(4.4.3)

Λj y¯rj − S¯r+ = y¯rj0 ; r = 1 , 2 , 3 , . . . s.

j=1

Λj ≥ 0; S¯i− ≥ 0; S¯r+ ≥ 0, ∀j, i, r. The objective function consists of the ratio of two linear components and hence called as linear Fraction Programming Problem (FPP). The given FPP can be transformed into LPP through Charnes Cooper transformation. An example of Charnes Cooper Transformation for radial model and non-radial model is given in the Appendix at the end of this thesis. Here, we directly write linear form of the model 4.4.3 as follows;

92

m

1 X s¯− i M inimize ρj0 = t − m i=1 x¯ij0 Subject to Constraints; s 1 X s¯+ r t+ =1 s r=1 y¯rj0 n X j=1 n X

λj x¯ij + s¯− xij0 ; i = 1 , 2 , 3 , . . . m. i = t¯

(4.4.4)

λj y¯rj − s¯+ yrj0 ; r = 1 , 2 , 3 , . . . s. r = t¯

j=1

λj ≥ 0; s¯− ¯+ r ≥ 0, ∀j, i, r; and t > 0. i ≥ 0; s The optimal solutions of the linear problem 4.4.4 can be substituted back to the model 4.4.3 to obtain the original solutions as follows; τj∗0 = ρ∗j0 ; Λ∗ =

+∗ λ∗ ¯−∗ s¯−∗ ¯+∗ = s¯ , S = and S t∗ t∗ t∗

(3.2.5)

The obtained values barSi−∗ and barSr+∗ represent the average input slacks and average output slacks over the study span. The only difference between the model 4.4.4 and Tones SBM model is that the former deal with the averages of all inputs and outputs whereas the Tone’s model deals with original values. Thus, the model provides an average efficiency measure for each DMU under study. The estimate of such efficiency scores provides a rough idea about how the DMU have performed during the period of time. Apart from its biased measure, it can be used to evaluate the crude measure of efficiency over the time.

4.5

Data and Selection of Variable

As mentioned above, window analysis deals with cross sectional and time varying data to obtain the efficiency trend over time. Windows of certain width are formed on the principle of moving averages. As there is no theory or justification that underpins the definition of the window width, this paper employs a three-year window, which is consists with Charnes et al. (1984). Furthermore, in most of studies, win-

93

dow length has been taken either as three or four (Avkiran, 2004). If the number of years is more, then window length of four will be favorable to reduce the number of windows, otherwise three is preferable in most of cases. Table 4.5.1 illustrates the procedure of framing the windows with window width of three. From the table, the first window incorporates the years 2007, 2008 and 2009 because of window width of three periods. To form a second window, the earliest period from the first window is dropped and the next consecutive period to first window is included. Thus, in second window, year 2007 will be dropped and year 2010 will be added to the window. Subsequently in third window, years 2009, 2010 and 2011 will be assessed. The analysis is performed until window 8 analyses years 2014, 2015 and 2016. As DEA window analysis treats a DMU as different entity in each year, a three-year window with ten DMUs is equivalent to 30 DMUs. Subsequently, by applying 8, three-year window would considerably increase the number of observations of the sample to 240, providing a greater degree of freedom. Let N= number of firms, K= number of windows, w=width of windows and T= number of time periods, then the number of windows can be can be obtained as follows; k = T − w +1

(4.5.1)

From this formula we have; Number of windows = k = T − w + 1 = 10 − 3 + 1 = 8 Number of different firms = N ∗ w ∗ K = 10 ∗ 3 ∗ 8 = 240 Thus, there are 240 different data points to which SBM DEA model is applied to evaluate the efficiency scores.

Window-1 Window-2 Window-3 Window-4 Window-5 Window-6 Window-7 Window-8

Table 4.5.1: Windows Breakdown 2007 2008 2009 2008 2009 2010 2009 2010 2011 2010 2011 2012 2011 2012 2013 2012 2013 2014 2013 2014 2015 2014 2015

2016

The definition and measurement of inputs and outputs in various contexts remains a 94

continuous issue among researchers, for example in banking functions it is difficult to define inputs and outputs. On the other hand, in the context of production functions, inputs and outputs seems to be clearly well defined. Thus, for cement companies there are four main inputs which are mainly responsible for the production process namely as (i) Power, Fuel and Water charges, (ii) Raw Materials, (iii) Employees cost, and (iv) Miscellaneous expenses. The output of cement companies can be considered in terms of Net Sales. The data set used in this analysis was obtained from the database Centre for Monitoring Indian Economy (CMIE) and the annual reports of Indian cement companies for the period 10 years from 2007-2016. The descriptive statistics of the variables (four inputs and one output) are given in the Table 4.5.2. Table 4.5.2: Descriptive Statistics of Variables Power, etc Raw Mat. Emp. Cost Misc. Exp Net Sales Mean 1121.84 788.07 302.72 1830.99 5310.78 Median 765.84 531.74 220.58 1243.61 3613 Max. 4742.89 3550.88 1343.02 9525.61 23708.79 Min. 110.79 48.62 23.88 139.88 883.48 St. Dev. 997.83 687.59 267.35 1937.07 4645.89

4.5.1

Window Analysis

The data set comprises of top ten Indian Cement Companies in terms of Net sales. As we have reliable data, took out directly from CMIE, we excrete the risk that incomplete or biased data may garble the estimation measures. Here we employed the extended DEA window analysis in non-radial SBM form to examine the efficiency trend of cement companies over the period 2007-16. For empirical analysis we used MATLAB, where we solved one by one as many as linear programming problems as number of DMUs in each window. The results of the SMB-DEA efficiency scores during the period 2007-2016 are presented in two Tables, the first five were put in Table 4.5.3 and the last five in the Table 4.5.3. To the best of our knowledge, there is currently no study in the literature that has analyzed Indian cement companies through SBM window analysis approach. Therefore, the results presented in the table provide the valuable information on the trends in performance of studied Indian cement companies. The SBM model is 95

5. India cements

4. Shree Cements

3. Ambuja Cements

2. ACC LTD

1. Ultratech Cements

Table 4.5.3: Window Analysis of SBM Efficiency Scores WinMean 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Mean SD dows /win. 1 1.00 0.74 0.69 0.81 2 0.79 0.72 0.71 0.74 3 0.71 0.93 0.74 0.79 4 1.00 0.97 0.94 0.97 0.91 0.11 5 1.00 0.99 0.96 0.98 6 0.99 0.96 1.00 0.98 7 0.96 1.00 1.00 0.99 8 1.00 1.00 0.98 0.99 1 0.76 0.61 0.63 0.67 2 0.60 0.61 0.64 0.61 3 0.73 0.71 0.66 0.70 4 0.93 0.89 0.95 0.93 0.85 0.16 5 0.90 0.98 1.00 0.96 6 0.96 0.99 1.00 0.99 7 0.97 0.98 1.00 0.99 8 0.98 1.00 1.00 0.99 1 0.05 0.06 0.05 0.05 2 0.06 0.05 0.04 0.05 3 0.07 0.06 0.64 0.26 4 0.06 0.85 0.86 0.59 0.57 0.39 5 0.88 0.89 0.93 0.90 6 0.89 0.92 0.86 0.89 7 0.92 0.85 0.91 0.89 8 0.85 0.91 0.93 0.90 1 0.45 0.22 0.23 0.30 2 0.22 0.24 0.24 0.23 3 0.31 0.29 0.28 0.29 4 0.39 0.43 0.69 0.50 0.61 0.31 5 0.50 0.80 1.00 0.77 6 0.67 0.83 1.00 0.83 7 0.83 1.00 1.00 0.94 8 1.00 1.00 1.00 1.00 1 0.45 0.41 0.40 0.42 2 0.41 0.40 0.50 0.44 3 0.49 0.55 0.63 0.56 4 0.80 0.89 1.00 0.89 0.73 0.22 5 0.88 1.00 0.95 0.94 6 1.00 0.83 0.90 0.91 7 0.81 0.86 0.82 0.83 8 0.84 0.80 0.93 0.86

applied in eight windows with width of three-years. The average efficiency scores for each cement company are given in the column named as Mean. The column labeled as SD represent the standard deviation for the efficiency score during the entire period. Any company may have different scores in different windows. If efficiency

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10. OCL India

9. Birla Corp.

8. Ramco Cements

7. JK Cement

6. Prism cement

Table 4.5.4: Window Analysis of SBM Efficiency Scores WinMean 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Mean SD dows /win. 1 0.57 0.49 1.00 0.68 2 0.49 0.83 1.00 0.77 3 0.91 1.00 1.00 0.97 4 1.00 1.00 0.58 0.86 0.79 0.09 5 1.00 0.59 0.55 0.71 6 0.84 0.80 0.88 0.84 7 0.71 0.77 0.85 0.78 8 0.72 0.73 0.71 0.72 1 0.62 1.00 0.92 0.85 2 1.00 0.92 0.61 0.84 3 1.00 0.72 0.45 0.72 4 1.00 0.65 0.49 0.71 0.82 0.09 5 0.69 0.53 0.99 0.74 6 0.49 0.96 0.97 0.81 7 0.96 0.96 0.92 0.95 8 0.96 0.92 0.87 0.92 1 0.56 0.47 0.44 0.49 2 0.47 0.45 0.47 0.46 3 0.59 0.57 0.51 0.56 4 0.72 0.70 0.75 0.73 0.70 0.17 5 0.70 0.79 0.84 0.78 6 0.77 0.85 0.88 0.83 7 0.85 0.87 0.95 0.89 8 0.87 0.93 0.74 0.84 1 0.55 0.55 0.58 0.56 2 0.52 0.55 0.56 0.54 3 0.64 0.64 0.60 0.63 4 0.80 0.75 0.81 0.79 0.74 0.15 5 0.74 0.80 0.82 0.79 6 0.79 0.86 0.94 0.86 7 0.83 0.91 0.93 0.89 8 0.89 0.91 0.89 0.90 1 0.63 0.42 0.43 0.49 2 0.35 0.36 0.31 0.34 3 0.40 0.32 0.38 0.37 4 0.33 0.39 0.36 0.36 0.50 0.18 5 0.40 0.36 0.40 0.39 6 0.47 0.54 0.67 0.56 7 0.51 0.56 1.00 0.69 8 0.51 0.94 1.00 0.82

rating of any DMU is stable irrespective of its window, then the respective DMU is efficient, which is according to the observation of Cooper et al. (2004). As mentioned above, we have taken the window width of three years because Charnes et al. (1984) observed that window width of three or four years leads in

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the balance of informativeness and stability of the performance measures. Further window width with three will relative increase the number of comparisons a 3 year window width has been chosen in this paper. Tables 4.5.3 and 4.5.3 reveal that SBM efficiency scores for 10 cement companies, each company is represented in 8 windows with width of 3. Each column is named with the year of comparison. For example, in the case of Ultratech Cements, the SBM efficiency scores of this company in the first window are 100.0, 74.0 and 68.5 respectively for the years 2007, 2008 and 2009. Subsequently, in the second window, the efficiency scores of 78.6, 71.9 and 70.7 correspond to years 2008, 2009 and 2010 respectively. In doing so, the table contributes itself to a study of trends. Two separate trends are revealed within windows by the adoption column view and column view. Observing the scores of the Ultratech Cement Company again in the second window, the efficiency score varies from 78 % to 70% over the years 2008 to 2010 by adopting of row view perspective. On the other hand, its efficiency also varies among the windows by adopting column view perspective. This variation excogitates simultaneously both the absolute performance of a cement company over time and relative performance of that company in comparison to its peers. The results of Tables 4.5.3 and 4.5.3 also revealed that the studied group of companies has exhibited mean SBM efficiency score of 72.2 per cent over the period 2007-2016, suggesting that the companies have performed relatively bad in their production process. It is clear from the table that Ultratech Cement was the best performer during the period with the mean efficiency score of 90.6 percent and accompanied by a relatively low standard deviation of 0.10. In contrast, our findings reveal that OCL India was the worst performer with 50.1 percent mean SBM efficiency with a standard deviation of 0.175. We noticed from the results that all the companies exhibited betterment and upward trend during the later part of studies. The Figure 4.2 indicates the upward trend in scores in the later part of study period. Figure 4.2 represents the average efficiency change over eight windows for each company. It can be revealed from the figure that in general the cement companies have experienced an increase in efficiency from 2007 to 2016. The first half of windows experiences a very high standard deviation while it is relatively very low in the later half. The Ultratech cements and ACC efficiency curves dominate the rest of companies.

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Figure 4.2: Efficiency Trend over windows

4.5.2

Average SBM Measure

The model 4.4.4 will be employed to each DMU to evaluate the average efficiency measurement. Before the models are to be executed, the original data set needs to transform into the data of averages for all inputs and outputs over the defined period of time. The Average SBM models are applied iteratively to evaluate the required estimates. We solved the models for each DMU and the results are presented in the Table 4.5.5. Table 4.5.5: Average SBM Scores through Different Methods DMUS

Avg.SBM

Ultratech cements ACC Ambuja Cements Shree Cements India cements Prism cement JK cement The Ramco Cements Birla Corporation OCL India

1 0.929 0.752 0.95 0.92 0.959 0.863 0.93 0.868 0.793

Avg. Ranking Ranking Window Analysis (Avg.SBM) (Window Analysis) 0.906 1 1 0.854 5 2 0.567 10 9 0.609 3 8 0.731 6 6 0.792 2 4 0.816 8 3 0.697 4 7 0.744 7 5 0.501 9 10

The results evaluated from the model 4.4.4 are presented in column second of the Table 4.4. The scores reveals that only DMU first (Ultratech) is average SBM efficient DMU. In other words, it can be said that during the span of ten years Ultratech cements dominates the frontier. We also compared the average SBM results 99

with the averages of SBM window analysis. The average efficiency scores of SBM window analysis are provided in the column three of the Table 4.5.5. The companies were ranked based on both the scores of DEA window analysis and Average SBM method. Columns five and six represents the ranks based on two approaches. The Ultratech Company was at the top from both the approaches. Two companies Ambuja and OCL got the last ranks from two approaches respectively. The companies Ultratech cements and India cements were ranked as first and sixth from both the approaches. Further, on comparing the two average score obtained through different methods, it is notices that the window analysis gives better results than the average SBM models. Consider the DMU first which cannot optimize the inputs and outputs as its efficiency score is one, while there is scope for optimization when considers window analysis approach. The comparison clearly identified from the Figure 4.3.

Figure 4.3: Comparison of Two Approaches Figure 4.3 clearly indicates the differences between the two approaches. The x-axis represents the different companies and y-axis represents the efficiency scores. The average SBM curve is everywhere above the window analysis curve, which indicates that the average SBM method lacks in identifying the full inefficiency sources. Thus, window analysis seems to better approach in dealing with time varying and cross sectional data.

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4.6

Conclusion

The chapter attempts to investigate the efficiency of top ten Indian cement companies during the period 2007-2016 through DEA window analysis. The preferred non-parametric DEA window analysis in SBM form have allowed us evaluated different efficiency scores. Further, the method is very useful in small sample size as it allowed greater degree of freedom. Additionally, we employed average SBM approach to calculate the average efficiency score over the studied period. During the study period, we observed an increasing of efficiency scores trend for all companies over the defined period. The first half of the study period experienced a low score of efficiency with very high standard deviation, whereas, in the later half, all the companies are closer to frontier. The company Ultratech cements, was most efficient among all of the companies. We also evaluated average SBM score of all companies over the period of ten years. The estimate will provide an approximate measure of efficiency score. Like window analysis, average SBM analysis reveals that Ultratech cements determines the frontier for all other companies. On the comparison of two approaches, it was observed that average SBM lacks in identifying the complete sources of inefficiency. Additionally, it does not reveal the efficiency change over time, but simply provides an approximate measure of efficiency. Further, the overall average efficiency score of all the companies in all the ten years is 72.2 percent, which is relatively a low efficiency score.

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Chapter 5

Multi-Period Performance Evaluation of Indian Commercial Banks Through DEA and MPI 5.1

Introduction

Productivity and efficiency of an organization are interconnected. However, efficiency is static as it does not consider the time for production whereas productivity is based on time. DEA Window Analysis is a suitable approach to estimate the efficiency change over time as described in chapter four. However, it neglects the effect of technological change over time. It assigns any technological change as of technical efficiency change. There are several methods to that could be used to evaluate productivity change, which includes Fisher index, Tornqvist index and the Malmquist Index. Among the three, Malmquist Total Factor Productivity (TFP) index is used most often to evaluate productive change as it has many advantages over the others. Grifell-Tatje and Lovell (1996) observed that there are three main advantages of Malmquist index relative to Tornqvist index and Fisher index. First, it does not require the prices of inputs and outputs which enable it to be used in non-profit organizations. Secondly, it does not require any assumption regarding the cost minimization or profit maximization. Finally, if one deals with panel data, it allows the decomposition of Productivity Changes (PC) into two mutually exclu102

sive components, i.e. Technical Efficiency Change (TEC) and Technological Change (TC). TEC and TC can be commonly referred as catching up and innovation respectively. Its primary drawback is that it is necessity to compute cumbersome distance functions for deriving the indices. However, DEA has made it easier to compute distance functions through LPPs. DEA-based Malmquist Productivity Index (MPI) thus differentiates the productivity change into two mutually exclusive and exhaustive components, namely, TEC and TC. TEC is defined as the score related to best practice frontier, whereas, the TC is defined as the shift of best practice frontier from the time t to t+1. This chapter is based on employing DEA to various Indian commercial banks for the specific year 2016 to evaluate different efficiency measurements and MPI to evaluate the productivity change, technical efficiency change and technological change of Indian commercial banks during the period 2012 to 2016. Indian banking system is classified into commercial banks and cooperate banks, where the former comprises of 95 percent of the banking system assets. Commercial banks are further categorized into 1) Public sector banks, comprises of SBI and its associates along with nationalized banks declared by Indian government in 1969 and onwards; 2) private sector banks comprises of old and new private sector banks; 3) regional rural banks; and 4) foreign banks. Due to growing competition among these banks it is necessary to create efficient, productive and profitable financial services by them. The efficiency of commercial banks has been widely and extensively studied in the last few decades. For commercial banks, efficiency implies improved profitability, greater amount of funds channelled in, better prices and services quality for consumers and greater safety in terms of improved capital buffer in absorbing risk (Berger and Humphrey, 1992). In India, the landscape of financial institutions has changed significantly with various liberalization measures being introduced in 1991. These includes government reforms to improve the bank infrastructure, existing ownership structures, lending practices and capital requirements, deregulation to allow for increased competition, focus on consolidation, mergers, acquisitions, etc. However, the impact of competition and regulatory changes could be judged by gross measures of performance such as profitability and failure rates. Economists and other financial experts are also interested in how such changes affect the efficiency with which banks transform the resources into financial services. This is because that the commercial banks have been facing an increasing degree of competition in the intermediation process from term lending institutions, nonbanking intermediaries (like mutual funds and leasing companies), chit funds and 103

the capital market. Besides this, new banking services like ATM machines and Internet banking are significantly growing due to the advancement of computers and information technology. The banks are facing pricing pressure, squeeze on spread and have to give thrust on retail assets. With the ongoing financial meltdown, the position of Indian banking sector has become more critical. In particular, the recent financial crisis has redefined the broad contours of regulation of the banking sector globally. This in turn has made it necessary to look for efficiency scores in the banking business. The banking efficiency scores, Evaluated through non-parametric Data Envelopment analysis (DEA) has attained a lot of interest in last two to three decades. This chapter is unfolded as follows. We begin with the brief review. Works related to estimating productivity change in commercial banks are discussed in section 5.2. Methodology of CCR and BCC models along with scale efficiency were discussed in section 5.3. This section also includes super efficiency model and numerical illustration of Indian commercial banks. Section 5.4 is devoted to present the models for productivity change, efficiency change and technological change and numerical illustration based on Indian banks. The final section 5.5 focused on the conclusions and research findings of the chapter.

5.2

Reported Research Work in DEA with MPI

The innovative work done by Charnes et al. (1978) for proposing CCR model is with CRS assumptions based on Farrell (1957). Later it was extended by Banker et al. (1984) to BCC by relaxing the assumption of CRS to include VRS. These two fundamental models have attracted much attention of researchers to examine various organizations. DEA has been extensively used in estimating efficiency in banking sector, because of variety of reasons. A measure of relative efficiency provides a good indicator of the success or failure of a bank in competitive market. Studies reveal that banks which operate efficiently have a better chance of sustaining their business in the future also. Berger and Humphrey (1992) found that during the 1980s, the high-cost banks experienced a higher rate of failure than more efficient banks. Wheelock and Wilson (1995) found that technical efficiency and likelihood of failure are inversely proportional. The lesser the technical efficiency is the more the likelihood of failure. Schaffnit et al. (1997) employed DEA methodology to

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evaluate bank branch allocative efficiency of large Canadian banks. Seiford and Zhu (1999) measured the efficiency of top 55 commercial banks via two-stage process measuring profitability and marketability respectively. In recent decades, all over the world DEA has been widely used as performance measurement tool in the banking industry. The other similar works are mostly as reported in Färe et al. (2004); Liang et al. (2008); Fukuyama and Weber (2009a,b, 2010); Halkos and Tzeremes (2013); Paradi et al. (2012). There are number of studies on technical/cost efficiency of banks in the context of Indian banking system. Bhattacharyya et al. (1997) examined the productivity efficiency of 70 Indian commercial banks through DEA and SFA during 1986-1991.Saha and Ravisankar (2000) evaluated the relative performance of public sector banks in India and found that the banks have in general improved their efficiency during 1992 to 1995. Sathye (2003) evaluated the efficiency of Indian banks industry in three groups that is publicly owned, privately owned and foreign owned banks. Their results reveal that there is no significant impact of entry of foreign banks in the field. Further, they obtained the limits (Sensitivity Analysis) of inputs and outputs so that current efficiency scores maintained.Debasish (2006) examined Indian banks through DEA for the period of 1997-2004 and observed that new banks are more efficient than older one during the period. Kumar and Gulati (2008) evaluated the extent of technical efficiency of 27 Indian public sector banks to provide strict ranking to these banks through super efficiency scores. Ray and Das (2010) evaluated the cost and profit efficiency of various Indian banks through DEA during the post reforms period and noticed that efficiency differentials are mostly affected by ownership of banks. Suzuki (2011) revealed whether the ownership of banks affect the banking performance in the context of Indian banks. Their results reveal that the foreign banks were ahead of all in terms of efficiency ratings during 2002 to 2009. Gulati et al. (2017) presented a holistic approach measuring overall efficiency and its decomposition in intermediation and operating efficiency. Further, they employed bootstrapped truncated regression algorithm to explore the influential determinants of intermediation and operating efficiency. In relation to the measurement of productivity changes over time of concerned DMUs, many studies have utilized MPI to evaluate such changes. Its literature can be presented in two ways. One is theoretical and other as empirical. In the former one, Caves et al. (1982a) introduced the concept of MPI which is based on the idea of Professor Sten Malmquist and was named after him. The MPI examines the relative 105

performance of a set of DMUs at different periods of time using technology of base period by computing different distance functions. On the other hand, Farrell (1957) solved the distance functions through LPPs.Fare et al. (1994) take the opportunity to formulate DEA-based MPI by combining the efficiency measurement of Farrell (1957) with the productivity measurement of Caves et al. (1982a) . Further, they decomposed the overall productivity into two mutually exclusive and exhaustive components, one of which measures the technical efficiency changes and the other frontier shift. In a similar pattern, Chen (2003) introduced non-radial MPI with an illustrative example of Chinese major industries. These MPI models were further discussed by Chen and Ali (2004) to get more insights on the second component of MPI by identifying the strategy shifts of individual DMUs based upon the changes of isoquant. They illustrated their proposed models with a set of Fortune Global 500 Computer and Office Equipment companies from 1991 to 1997. Pastor and Lovell (2005) proposed a model for obtaining global MPI score of same Global 500 Computer and Office Equipment companies from 1991 to 1997. In the empirical context, there are number of studies on examining productivity change through DEA based MPI. Elyasiani and Mehdian (1990) derived the efficiency measures and technological change over time for a sample of large US commercial banks during 1980 to 1985. Their results reveals that the banks possess high pace of technological advancement during the period. Fukuyama and Weber (2002) estimated the output allocative efficiency and productive change of Japanese banks over the period from 1992-1996. They conveyed that the banks could have used only 78% to 93% of the current inputs, if they had chosen the revenue maximizing output mix. Rezvanian et al. (2008) observed the effect on productivity growth, efficiency change and technological progress due to ownership effects for Indian banking industry during 1998 to 2003. Fung et al. (2008) examined the efficiency and productivity changes through DEA-MPI in regional airports in China during the years 1995 to 2004. Barros and Weber (2009) evaluated the productivity of UK airports during 2000-2005. Feng and Serletis (2010) examined the US large banks during 2000 to 2005 to evaluate their estimates in total factor productivity growth, efficiency change and technological change along with economies of scale. Wang and Lan (2011) measured MPI based on double frontiers DEA and applied the proposed models to the productivity analysis of the industrial economy of China. Kao and Hwang (2014) evaluated multi-period efficiency and MPI in two-stage production process 106

of Taiwanese non-life insurance companies. Thanassoulis et al. (2015) developed an index for comparing the productivity in terms of cost with known input costs. ¨ u et al. (2016) evaluated the efficiency of Turkish airports over the period 2009Orkc¨ 14 using DEA and Malmquist productivity index. They decomposed the overall productivity growth over time into efficiency growth and technical growth. In recent times, much attention has been taken on productivity growth. Kao (2017) measured and decomposed the MPI in case of parallel stage production systems of various Iranian commercial banks. Badunenko and Kumbhakar (2017) proposed a model where it controls bank-heterogeneity, and they have introduced persistent and timevarying inefficiency measures. Additionally they incorporated determinants of both persistent and time-varying inefficiency as well as production risks.

5.3

Data Envelopment Analysis and its Testing

Prior to applying the conventional proposed DEA models to real life problems, this section includes the underlying concepts of DEA formulation.

5.3.1

DEA Concept

Productivity of a particular DMU can be evaluated from the ratio output/input. This measure is not enough to compare it with other DMU producing same output by using same input. For comparison, it is necessary to estimate the efficiency of DMUs. This measure also does not work well when the DMUs possess multiple inputs and multiple outputs. DEA was acquainted as a way to calculate the performance measure of DMU using various inputs to produce various outputs. It calculates the measure of performance through linear programming technique. The basic DEA model known as CCR model is based on the CRS assumptions. Later on the BCC model came in to existence to relax the CRS by introducing Variable Returns to Scale (VRS). These two basic DEA models can identify the frontier with benchmark entities (DMUs) and suggest the degree of improvement needed for inefficient DMUs to be efficient. This improvement could be either by minimizing the inputs or to maximizing the outputs by the obtained degree depends upon the orientations considered.

107

5.3.2

Charnes Cooper Rhodes (CCR) Model

Consider the set of n DMUs transforming m number of inputs to s number of outputs. As per usual notation xij , (i = 1, · · · , m) and yrj , (r = 1, · · · , s) respectively represents the ith input and rth output of the j th , (j = 1, · · · , n) DMU. The objective of CCR model is to maximize the ratio of virtual set of outputs and virtual set of inputs of a particular DMU under study, with the restriction that the ratio should be less than or equal to unity for all the DMUs under study. In this study jo , (jo ∈ J) is denoted as the evaluated DMU and Ejo is the corresponding efficiency measure. Mathematically, the problem can be formulated as follows; Ps ur yrjjo M aximize Ejo = Pr=1 m i=1 vi xijjo Subject to Constraints; Ps ur yrj Pr=1 ≤ 1, ∀ j = 1, · · · , n m i=1 vi xij ur , vi ≥ ε;

(5.3.1)

∀ i, r

Here, the weights of inputs and outputs are assumed to be strictly positive. The variable ε is very small positive number. The model 5.3.1 represents the ratio of two LPPs and hence called as FPP. This FPP could have infinite number of solutions. The model can be transformed into linear one to get a unique optimal solution in a scalar form through Charnes-Cooper transformation. The transformed model is given as follows;

M aximize Ejo =

s X

µr yrjjo

r=1

Subject to Constraints; m X νi xijjo = 1

(5.3.2)

i=1 s X

µr yrj −

r=1

µr , νi ≥ ε;

m X

νi xij ≤ 0, ∀ j = 1, · · · , n

i=1

∀ i, r

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The set of constraints contains a constraint for each of the DMUs under study. If the number of DMUs is large, then solving the problem could be more complex. In contrast, the dual model’s constraint set depends on the number of inputs and outputs used and hence will comprise relatively very less number of constraints. Furthermore, the weights are assigned to inputs and outputs rather than DMUs and make it difficult to get a reference DMU for inefficient DMUs. Thus, it looks feasible to deal with dual model rather than the original for efficiency evaluation. M inimize θjo − ε

X m i=1

s− i

+

s X

s+ r

r=1

Subject to Constraints; n X λj xij + s− i = θjo xijo ;

(5.3.3) i = 1, · · · , m

j=1 n X


r = 1, · · · , s

j=1 + λj , s− i , sr ≥ 0, ∀ j, i, r

The model 5.3.3 is solved for all DMUs to obtain their corresponding efficiency scores. This model is generally used when all the DMUs are on optimal scale; hence the efficiency obtained from model 5.3.3 is a combination of technical efficiency and scale efficiency. The following BCC model discriminates between the technical efficiency and scale efficiency.

5.3.3

Banker Charnes Cooper (CCR) Model

The BCC model is the extension of CCR model to evaluate pure technical efficiency based on the assumption of VRS. This method excludes the scale effect and becomes the reason of being always greater than or equal to CCR efficiency1 . The BCC model in additive form can be formulated by introducing the amount of 1

If a DMU is CCR efficient, then it is necessarily BCC efficient, while the converse is not true.

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scale efficiency in the objective function as well as in constraints as follows; M aximize Ejo =

s X

µr yrjjo + ω

r=1


(5.3.4)

i=1 s X

µr yrj −

r=1

m X

νi xij + ω ≤ 0, ∀ j = 1, · · · , n

i=1

µr , νi ≥ ε;

∀ i, r;

ω is unrestricted.

The Model 5.3.4 evaluates the efficiency score which enables to determine the returns to scale of DMUs. If ω > 0 then the corresponding DMU possess Increasing Returns to Scale (IRS), and if ω < 0, then the corresponding DMU possess Decreasing Returns to Scale (DRS) and if it is equal to one ω = 0, the DMU possess the Constant Returns to scale (CRS). A DMU is said to be Pareto-efficient if and only if it is pure technical efficient. The dual or the envelopment form of the model 5.3.4 is as follows;

M inimize θjo − ε

X m i=1

s− i

+

s X

s+ r

r=1

Subject to Constraints; n X λj xij + s− i = θjo xijo ; j=1 n X j=1 n X


(5.3.5) i = 1, · · · , m r = 1, · · · , s

λj = 1

j=1 + λj , s− i , sr ≥ 0, ∀ j, i, r + Model 5.3.5 is called as input oriented BCC model, where the variables s− i and sr represent the input excesses and output shortfalls respectively. The optimal value of the model is always less than or equal to one. In a similar way, output oriented CCR and BCC models can be formulated. The output oriented CCR model in

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envelopment form is as follows; M aximize φjo + ε

X m

s− i

i=1

+

s X

s+ r

r=1

Subject to Constraints; n X λj xij + s− i = xijo ; j=1 n X j=1 n X

(5.3.5) i = 1, · · · , m

λj yrj − s+ r = φjo yrjo ;

r = 1, · · · , s

λj = 1

j=1 + λj , s− i , sr ≥ 0, ∀ j, i, r

The output oriented BCC model can be formulated by adding the convexity constraint to the model 5.3.6. The basic difference between the CCR and BCC model is that the former does not consider the scale efficiency. The CCR model is based on the assumption that all the DMUs are working on the optimal scale. The assumption may not be true in some real-life situations. The BBC model on the other side relaxes this assumption of scale efficiency.

5.3.4

Scale Efficiency

As mentioned above, CCR efficiency is the combination of pure technical efficiency and scale efficiency and BCC efficiency is only pure technical efficiency excluding scale efficiency. Thus, the radial difference between the CCR frontier and BCC frontier is called as scale efficiency. Thus, we have equation for scale efficiency as; Scale Ef f iciency =

∗ θCCR ∗ θBCC

(5.3.6)

As BCC score is always greater than or equal to CCR, scale efficiency score is always less than one. If both CCR and BCC scores are equal to one for a set of DMUs, then their scale efficiency are also equal to one and are called as Most Productive Scale Size (MPSS) DMUs. A ranking procedure cannot be done for the MPSS set of DMUs, as all of having same efficiency score and a tie situation, leads 111

to need of super efficiency scores.

5.3.5

DEA Model for Super-Efficiency

The CCR and BCC models examines the set of DMUs in any orientations whose performance score is equal to one. If the set is singleton set then ranking procedure could be done easily. However, if the set contains more than one efficient DMU, then it is impossible to rank them as all of them having same efficiency score of one. Andersen and Petersen (1993) developed a super efficiency model to discriminate among these efficient units. This model excludes the evaluated DMU in the constraint set which limits all the DMUs less than one. After excluding a DMU from the constraint set it will posses efficiency scores greater than one, which leads in easy way of ranking. In doing so, there is no effect on the efficiency of inefficient DMUs whose score will be always less than one. This can be observed in the following model for DMU jo under evaluation.

M aximize Ejo =

s X

µr yrjjo

r=1


(5.3.8)

i=1 s X

µr yrj −

r=1

µr , νi ≥ ε;

m X

νi xij ≤ 0, ∀ j, &j 6= jo (DM U under study)

i=1

∀ i, r

Applying model 5.3.8 to a set of efficient DMUs will provide super efficiency score of each DMU. Model for super efficiency can be formulated for all types of DEA models by removing the constraint of evaluated DMU from the set of constraints. A numerical illustration of Indian commercial banks is given in the succeeding section.

5.3.6

Variables and Analysis

For computing the relative efficiency scores, the most difficult task that analysts always encounter is to select the relevant inputs and outputs. Although, it seems

112

very easy to define inputs and outputs for production firms like Cement companies, steel companies and many other manufacturing companies. But in case of banks it will not be an easy task. In the literature on banking performance there are three approaches for selecting the input and output variables for a bank. Those are (i) Intermediation Approach, (ii) User Cost Approach and (iii) Value Added Approach. Most of the DEA studies follow intermediation approach, as it seems to be more suitable for evaluating the efficiency of banking sector. Therefore, in this study intermediation approach is used for selection of variables, which consider banks as financial intermediaries. The variables classified in our study are (i) Total Assets, (ii) Deposits and (iii) borrowings as input variables; whereas the classified output variables are (i) Operating Profit, (ii) Interest Income (Spread), (iii) Advances and (iv) Investments.

5.3.7

Data Collection and Processing Methodology

The data of inputs and outputs was taken from BLOOMBERG database. Initially, we have obtained the data of 45 banks, but due to missing values in the data and incompatibility of data we restricted it to 29 banks. We deleted the banks under study whose data was not available for the study period. We have calculated different efficiency scores by solving LPPs of all DMUs one by one in excel using solver and the results are provided in the Table 5.3.1. Table 5.3.1 recapitulates the DEA technical efficiency scores of 29 Indian commercial banks during the multi-year period of 2012 to 2016 using the CCR model. It is observed that the J&K Bank, South Indian Bank, Indian Bank, HDFC Bank and Kotak Mahindra Bank are all efficient and their rank is one for all years under study as per the DEA analysis. Bank of Baroda, PNB, DCB bank were on the top list among inefficient banks. The analysis shows that RBL bank has lowest efficiency among all banks and thus ranked as last. RBL bank got highest efficiency of 81 % in the year 2015. The year 2015 was most productive among all years as highest number of DMU got efficiency score as 1 compared to other years. Most of the banks during this period have achieved 90 to 95 percent efficiency level score, where as some get below 90 percent. The conventional DEA scores categorized the banks into efficient and non-efficient groups. All efficient DMUs in terms of average efficiency got rank one and rest were ranked according on the basis of their efficiency score. The above procedure of ranking has some ambiguity due to the tie in the technical 113

Table 5.3.1: CCR-Output Oriented Efficiency during 2012-16 Bank Name J&K B. South Indian. B. Indian B. HDFC B. Kotak Mahn. B. of Baroda PNB DCB B. ICICI B. Andhra B, Axis B. Corp B. Karur Vysya SBI IDBI B. UCO B. Federal B. Allahabad B. Union B.Ind. Karnataka B. Orien B.Com. B. of India Syndicate B. Ind. Overs. B. Dena B. Cntrl B. Ind. Canara B. Yes B. RBL B.

2012 1.000 1.000 1.000 1.000 1.000 1.000 0.954 0.938 0.905 1.000 0.925 0.974 0.927 0.984 0.985 0.976 0.965 0.963 1.000 0.870 0.913 0.990 0.949 0.921 0.932 0.917 0.887 0.850 0.705

2013 1.000 1.000 1.000 1.000 1.000 0.978 0.985 0.935 1.000 0.956 0.949 0.938 0.945 0.965 0.967 1.000 0.932 0.910 0.949 0.913 0.934 0.910 0.941 0.936 0.865 0.918 0.843 0.754 0.655

2014 1.000 1.000 1.000 1.000 1.000 1.000 0.942 1.000 0.966 0.911 0.974 1.000 0.972 0.932 0.943 0.956 0.903 0.934 0.873 0.911 0.931 0.904 0.917 0.886 0.913 0.846 0.850 0.870 0.693

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2015 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.986 1.000 0.964 0.959 1.000 1.000 1.000 0.961 1.000 0.949 0.960 0.955 0.911 0.952 0.877 0.898 0.929 0.819

2016 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.946 1.000 0.953 0.930 0.839 0.938 0.926 0.925 1.000 0.946 0.904 0.879 0.918 0.874 0.944 0.855 0.924 0.765

Average TE 1.000 1.000 1.000 1.000 1.000 0.995 0.976 0.974 0.974 0.973 0.969 0.968 0.968 0.959 0.956 0.954 0.947 0.946 0.941 0.938 0.934 0.933 0.928 0.914 0.907 0.900 0.866 0.865 0.727

Rank 1 1 1 1 1 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

efficiency among the benchmarking banks. In order to make more proper ranking, super efficiency is considered in the place of technical efficiency. The ranking of banks is carried out based on the records of all 5 years in the Table 5.3.2. Table 5.3.2: Super Efficiency of Banks during 2012-16 Bank Name Dena B. J&K B. South Ind. B. Syndicate B. Indian Overs. B. Andhra B, Karnataka B. UCO B. ICICI B. Orien B.Com. DCB B. Indian B. IDBI B. SBI B. of Baroda Union B.Ind. HDFC B. Corp B. Kotak Mahn. B. of India Canara B. PNB RBL B. Federal B. Allahabad B. Axis B. Cntrl B. Ind. Yes B. Karur Vysya

2012 2013 2014 2015 2016 Mean

CV

0.932 1.775 4.107 0.949 0.921 1.002 0.87 0.976 0.905 0.913 0.938 1.143 0.985 0.984 1.049 1.035 1.076 0.974 1.875 0.99 0.887 0.954 0.705 0.965 0.963 0.925 0.917 0.85 0.927

0.037 0.059 0.351 0.03 0.018 0.048 0.078 0.099 0.117 0.014 0.094 0.177 0.02 0.018 0.025 0.056 0.018 0.068 0.167 0.038 0.025 0.034 0.08 0.038 0.035 0.051 0.039 0.073 0.075

0.865 1.6 2.388 0.941 0.936 0.956 0.913 1.141 1.004 0.934 0.935 1.112 0.967 0.965 0.978 0.949 1.02 0.938 1.663 0.91 0.843 0.985 0.655 0.932 0.91 0.949 0.918 0.754 0.945

0.913 1.749 1.6 0.917 0.886 0.911 0.911 0.956 0.966 0.931 1.064 1.348 0.943 0.932 1.025 0.873 1.041 1.125 1.588 0.904 0.85 0.942 0.693 0.903 0.934 0.974 0.846 0.87 0.972

0.952 1.536 1.922 0.955 0.911 1.012 1.007 1.016 1.158 0.949 1.084 1.774 0.959 0.964 1.044 0.961 1.032 0.986 1.368 0.96 0.898 1.024 0.819 1.009 1.004 1.013 0.877 0.929 1.112

0.874 1.57 2.337 0.879 0.918 1.045 1.074 0.839 1.235 0.946 1.195 1.305 0.93 0.953 1.019 0.925 1.034 0.946 1.134 0.904 0.855 1.019 0.765 0.938 0.926 1.068 0.944 0.924 1.084

0.907 1.646 2.471 0.928 0.914 0.985 0.955 0.986 1.054 0.935 1.043 1.336 0.957 0.96 1.023 0.949 1.041 0.994 1.526 0.934 0.867 0.985 0.727 0.949 0.947 0.986 0.9 0.865 1.008

Mean based CV based Ranking Ranking 25 11 2 18 1 29 23 8 24 3 13 15 17 22 12 25 5 26 21 1 6 24 4 28 16 5 15 2 8 6 19 17 7 4 10 19 3 27 22 13 27 7 14 9 29 23 18 12 20 10 11 16 26 14 28 20 9 21

Table 5.3.2 recapitulates the technical efficiency of inefficient banks and super efficiency of efficient banks. We have calculated Mean and Coefficient of variation (CV) of every bank based on five year period. Since the super efficiency of inefficient DMUs are same as their usual technical efficiency so it is necessary to calculate the super efficiency only for efficient DMUs. Analysis is focused with two types of rankings; one is based on the mean and the other is on coefficient of variation. The mean efficiency provides score based on the efficiency scores of all the years under study. The highest average efficiency is treated as first rank. The case is reverse in 115

case of CV, where the bank having least CV will have the rank one. This CV shows the ups and downs of efficiency during five years. The bank having efficiency one in terms of CV shows that all the five years having more or less the same efficiency. The last ranked bank in CV scenario, shows that the banks has high fluctuations in efficiency scores for different years. We can also calculate other efficiency scores like CCR, BCC and scale efficiency in both input and output orientations for the specific year 2016. CCR efficiency is mixed up of pure technical efficiency and scale efficiency, where as the BCC model provides pure technical efficiency. The radial difference between CCR frontier and BCC frontier is scale efficiency. Table 5.3.3 summarizes all the efficiency scores in both orientations along with returns to scale. Table 5.3.3: CCR and BCC efficiency scores based on year 2016 Input Oriented Banks Dena B. J&K B. South Ind. B. Syndicate B. Ind. Overs. B. Andhra B, Karnataka B. UCO B. ICICI B. OrienB.Com. DCB B. Indian B. IDBI B. SBI B. of Baroda Union B.Ind. HDFC B. Corp B. Kotak Mahn. B. of India Canara B. PNB RBL B. Federal B. Allahabad B. Axis B. Cntrl B. Ind. Yes B. Karur Vysya

Output Oriented

CCR-I BCC-I Scale RTS CCR-O BCC-O Scale RTS 0.874 1.000 1.000 0.879 0.918 1.000 1.000 0.839 1.000 0.946 1.000 1.000 0.930 0.953 1.000 0.925 1.000 0.947 1.000 0.904 0.855 1.000 0.765 0.938 0.926 1.000 0.944 0.924 1.000

0.879 1.000 1.000 0.985 0.921 1.000 1.000 0.859 1.000 1.000 1.000 1.000 0.934 1.000 1.000 0.967 1.000 1.000 1.000 1.000 1.000 1.000 0.802 0.971 0.977 1.000 1.000 0.933 1.000

0.994 1.000 1.000 0.892 0.997 1.000 1.000 0.977 1.000 0.946 1.000 1.000 0.996 0.953 1.000 0.957 1.000 0.947 1.000 0.904 0.855 1.000 0.955 0.965 0.948 1.000 0.944 0.991 1.000

IRS CRS CRS IRS IRS CRS CRS IRS CRS IRS CRS CRS IRS IRS CRS IRS CRS IRS CRS IRS IRS CRS IRS IRS IRS CRS IRS IRS CRS

116

0.874 1.000 1.000 0.879 0.918 1.000 1.000 0.839 1.000 0.946 1.000 1.000 0.930 0.953 1.000 0.925 1.000 0.947 1.000 0.904 0.855 1.000 0.765 0.938 0.926 1.000 0.944 0.924 1.000

0.876 1.000 1.000 0.987 0.921 1.000 1.000 0.861 1.000 1.000 1.000 1.000 0.939 1.000 1.000 0.968 1.000 1.000 1.000 1.000 1.000 1.000 0.792 0.975 0.979 1.000 1.000 0.931 1.000

0.997 1.000 1.000 0.891 0.997 1.000 1.000 0.975 1.000 0.946 1.000 1.000 0.990 0.953 1.000 0.956 1.000 0.947 1.000 0.904 0.855 1.000 0.966 0.962 0.946 1.000 0.944 0.992 1.000

IRS CRS CRS IRS IRS CRS CRS IRS CRS IRS CRS CRS IRS IRS CRS IRS CRS IRS CRS IRS IRS CRS IRS IRS IRS CRS IRS IRS CRS

Table 5.3.3 exhibits various efficiency scores to determine recent performances through various perspectives. The table summarizes the the scores evaluated through CCR and BCC model in both input and output orientation. Additionally, we computed scale efficiency and according determined the returns to scale in both orientations. It can be observed from the analysis that scale efficiency reaches 1.0, when the Bank has the constant returns to scale. The CCR scores in both input and output orientations reveals the banks which are efficient are: J&K Bank, South Indian Bank, Andhra bank, Karnataka bank, ICICI Bank, DCB Bank, Indian Bank, Bank of Baroda, HDFC Bank, Kotak Mahindra Bank, PNB, Axis Bank and Karur Vysya Bank. These banks are not only CCR input and output oriented efficient, but they are also Most Productive Scale Size (MPSS) with a sore 1.0 in CCR, BCC and Scale efficiency. Herein, the BCC efficiency score which actually represents pure technical inefficiency can be increased by proper combinations of inputs and outputs. On the other hand, scale efficiency can be increased by increasing the capacity of the bank.

5.4

Malmquist Productivity Index (MPI)

Turning to measurement of productivity change over time, Malmquist Productivity Index is one of the popular methods. Malmquist (1953) initial work remains unnoticed and inapplicable until (Caves et al., 1982a) reintroduced it to productivity measurement. Subsequently, Färe et al. (1992) elaborated the approach. Their major extension of the index was its decomposition into a measure capturing efficiency change and other capturing technical change over time. The first component of efficiency change measures the shift of individual DMU in comparison to its frontier and the second component of technical change measures shift in frontier over time. Following their work, the approach became a standard methodology to evaluate the productivity change, efficiency change and technical change between any two periods. The concept is well illustrated in the Figure 5.1, which presents a case of single input single output with CRS assumption. Lets assume for each time period t = 1, · · · , T, the production technology S t modelled the transformation of inputs xt ∈ 1) and any impairment in performance yields Malmquist index less than unity (Mj < 1) . At times it may happen that efficiency change and technical change are moving in opposite directions. For example, a Malmquist productivity index greater than unity say, 1.2 which indicates productivity gain, could have efficiency change less than unity say 0.8 and technical change greater than unity 1.5, makes the ambiguity in conclusions. To sum up this (Färe et al., 1992) defined productivity growth as intersection of efficiency change and technical change. They interpreted the components of productivity growth as: ”im120

provements in efficiency change are considered to be catching up while improvements in technical change are considered to be evidence of innovation”. This decomposition thus provides a way for testing the source of change in productivity. To know the source it is necessary to measure the Malmquist productivity index. Various approaches are known to calculate this index. Herein we apply linear programming approach because of its various advantages over the other methods. Suppose we have usual notations of inputs and outputs for n DMUs in T time periods. The reference technology for the time period t could be as follows; n S t = xt , y t : xtt ≥

n X

(5.4.10)

λtj xtij

i = 1, · · · , m

j=1

yrt ≤

n X

t λtj yrj

r = 1, · · · , s

j=1

λtj ≥ 0, ∀ j,

o

P This model exhibits CRS, which can be relaxed by adding the constraint nj=1 λtj = 1 to include VRS. In order to calculate MPI of a DMU jo between time period t and ’t+1’, we need to solve LPPs of four input distance functions Djto xt , y t , t+1 t+1 t+1 t t . Now, to obtain these distance and D x , y x , y Djto xt+1 , y t+1 , Djt+1 j o o functions it is assumed that input distance functions are reciprocal of input oriented technical efficiency. Assuming CRS with input orientation, the distance function of DMU jo in time t referring to frontier S t is as;

−1 t t t Djo x , y

  M inimize θjo      to Constraints;  Subject Pn t t t = j=1 λj xij ≤ θjo xijo , i = 1, · · · , m  P  n t t t  r = 1, · · · , s  j=1 λj yrj ≥ yrjo ,    t λj ≥ 0, ∀j = 1, ..., n

(5.4.11)

Similarly, the LPP for distance function of DMU jo (jo ∈ J) in time ’t+1’ relative

121

to production frontier S t is

−1 t t+1 t+1 Djo x , y

  M inimize θjo      to Constraints;  Subject Pn t+1 t t = j=1 λj xij ≤ θjo xijo , i = 1, · · · , m  Pn  t+1 t t  r = 1, · · · , s  j=1 λj yrj ≥ yrjo ,    t λj ≥ 0, ∀j = 1, ..., n

(5.4.12)

The LPP equivalent to the distance function for DMU jo in time t relative to production frontier S t+1 is

−1 t+1 t t Djo x , y

  M inimize θjo (5.4.13)      to Constraints;  Subject Pn t+1 t+1 t = j=1 λj xij ≤ θjo xijo , i = 1, · · · , m  P  n t+1 t+1 t  r = 1, · · · , s  j=1 λj yrj ≥ yrjo ,    t+1 λj ≥ 0, ∀j = 1, ..., n

The last and the forth distance function equivalent LPP for the DMU jo at time ’t+1’ with the reference technology of S t+1 is

−1 t+1 t+1 t+1 D jo x , y

  M inimize θjo (5.4.14)      to Constraints;  Subject Pn t+1 t+1 t+1 = j=1 λj xij ≤ θjo xijo , i = 1, · · · , m  Pn  t+1 t+1 t+1  r = 1, · · · , s  j=1 λj yrj ≥ yrjo ,    t+1 λj ≥ 0, ∀j = 1, ..., n

The first two distance among four are with respect to reference technology t and the later two with respect to reference technology ’t+1’. Since the distance function Djto xt+1 , y t+1 is obtained with reference to time t+1 which may be outside the convex set pertaining to t hence it is always greater than or equal to one. Thus we need to solve above four linear programming problems 5.4.11 to 5.4.14 to get the estimates of the productive change, efficiency change and technological change between the two time periods by substituting the optimal values in expression 5.4.8. It should be noted that for allowing VRS to further decompose the efficiency in the MPI expression 5.4.8 into pure technical efficiency change and scale efficiency P change, an additional constraint nj=1 λj = 1 should be added to two LPPs in 5.4.11 122

and 5.4.14. Following Färe et al. (1989) the scale efficiency change (SE) can be evaluated as follows; Dt+1 xt+1 , y t+1 ) SE t+1 xt+1 , y t+1 ) jo × × Mjo xt+1 , y t+1 , xt , y t = V RS DVt RS xt , y t ) SEjto xt , y t ) " ! !#1/2 t t DCRS xt+1 , y t+1 ) DCRS xt , y t ) t+1 t+1 DCRS xt+1 , y t+1 ) DCRS xt , y t )

(5.4.15)

Where Scale efficiency Change = SEjt+1 xt+1 , y t+1 ) Scale ef f iciency at (0 t + 10 ) o = Scale ef f iciency at 0 t0 ) SEjto xt , y t )

(5.4.16)

Pure technical efficiency change= t+1 t+1 DVt+1 ,y ) P ure technical ef f iciency at (0 t + 10 ) RS x = t 0 0 t P ure technical ef f iciency at t DV RS x , y t )

(5.4.17)

The values of 5.4.16 and 5.4.17 can be easily obtained by their respective LPPs. A numerical illustration of Indian commercial banks by employing the MPI is given in the next section.

5.4.1

Analysis of Malmquist Productivity Index

As mentioned above, the advantage of DEA-based MPI to split the total productivity into two different measures leads to good managerial decisions. This helps in not only helps in monitoring the performance trends but also observes the effect of innovation on the production process. This section presents the total productivity change with its two two decomposed component of sample of Indian commercial banks during the five-year span from 2012 to 2016. The outline of bank productivity change for the five year period is shown in Figure 5.2. Figure 5.2 show that all the three changes namely productivity change, efficiency change and technical change decline between 2012 and 2013. Out of three changes it is observed that technical change has faster declining than other two. It can be seen from the figure that productivity change and efficiency change are having more or

123

Figure 5.2: Productivity, Efficiency and technological changes less the same trend except 2012-13. Technical change declines in initial phase then steadily increasing between 2014 and 2015 and rises faster to reach productivity change in 2015-16. Bank wise changes for all the three are given in Table 5.4.1. Table 5.4.1 is also providing the productivity change in addition to all the three changes. The first part of the Table 5.4.1 recapitulates the results of Productivity change, Efficiency change and Technical change between 2012 and 2016. It is evitable from table that total productivity change is influenced much heavily by technical change than efficiency change. It should be noted that MPI2 greater than one means increase of total productivity, less than one means decrease and equal to one means no change in total productivity between the respective periods. DCB bank got highest total productivity change between the years followed by Union Bank of India and Syndicate bank. The only bank with regress in productivity is Kotak Mahindra bank. In case of efficiency change it is Karnataka Bank which got highest efficiency change value 1.149. There are many banks whose efficiency change index is less than one hence indicates regress in efficiency change between the periods 2012 and 2016. The lowest one among them is UCO bank. For technical change, DCB bank ranks the first followed by union bank of India. The lowest with index less than one is only Kotak Mahindra bank. 2

Note: Index > 1 suggests a gain in total productivity, Index < 1 suggests a fall in total productivity and Index = 1 suggests the constant productivity

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Table 5.4.1: MPI changes during 2012-16 & Productive changes over years Banks Dena B. J&K B. South Ind. B. Syndicate B. Ind.Overs. B. Andhra B, Karnataka B. UCO B. ICICI B. OrienB.Com. DCB B. Indian B. IDBI B. SBI B. of Baroda Union B.Ind. HDFC B. Corp B. Kotak Mahn. B. of India Canara B. PNB RBL B. Federal B. Allahabad B. Axis B. Cntrl B. Ind. Yes B. Karur Vysya

Productivity Efficiency Technical Change Change Change 2012-16 1.391 1.066 1.389 1.432 1.403 1.343 1.418 1.402 1.083 1.393 1.678 1.253 1.338 1.201 1.322 1.435 1.134 1.158 0.886 1.428 1.407 1.303 1.341 1.225 1.143 1.189 1.429 1.216 1.347

2012-16 0.937 1.000 1.000 0.927 0.997 1.000 1.149 0.860 1.106 1.037 1.066 1.000 0.944 0.968 1.000 0.925 1.000 0.971 1.000 0.913 0.964 1.048 1.086 0.972 0.962 1.081 1.029 1.088 1.079

Productivity Change

2012-16 2012-13 2013-14 2014-15 2015-16 Average 1.935 0.983 1.067 1.029 1.097 1.044 1.136 1.330 0.795 1.026 1.000 1.038 1.929 0.644 0.793 1.185 1.000 0.906 2.051 0.973 0.997 1.010 1.092 1.018 1.970 1.154 0.915 0.962 1.094 1.031 1.803 1.022 0.924 1.025 1.000 0.993 2.010 1.094 0.953 1.195 1.000 1.061 1.966 1.193 0.931 1.028 1.092 1.061 1.174 1.058 0.990 1.033 1.000 1.020 1.941 1.018 0.964 1.021 1.055 1.015 2.817 1.061 1.107 1.001 1.000 1.042 1.570 1.217 0.873 1.244 1.000 1.084 1.790 1.079 0.961 0.953 1.059 1.013 1.443 0.996 0.984 0.994 1.044 1.005 1.748 0.914 1.050 0.999 1.000 0.991 2.059 1.008 0.932 1.021 1.061 1.006 1.285 1.033 1.017 0.984 1.000 1.009 1.342 0.977 1.132 0.937 1.035 1.020 0.785 1.021 1.027 0.899 1.000 0.987 2.038 1.052 1.005 1.025 1.074 1.039 1.979 0.995 1.005 1.000 1.141 1.035 1.697 1.095 0.967 1.021 1.000 1.021 1.798 1.044 1.032 1.038 1.263 1.094 1.500 1.005 0.985 1.077 1.033 1.025 1.305 0.983 1.012 1.058 1.039 1.023 1.414 1.035 0.995 1.016 1.000 1.012 2.042 1.122 0.915 0.941 1.099 1.019 1.478 1.021 1.060 0.999 1.080 1.040 1.814 1.076 1.033 1.039 1.000 1.037

The second part of the Table 5.4 recapitulates productivity change from year to year of five year period. The results shows that the change of total productivity over time, which decreased from 2012 to 2013 and increased from 2013 to 2014 and later on. Overall, bank productivity increased slightly from 2012 to 2016 by 2.4%. In particular, 25 banks increased their productivity, whereas 4 banks suffer from a decreased productivity. Indian Bank ranks the first, where productivity increased from 2012 to 2013 and dipped substantially in 2013-2014. There was no change in productivity in 2014-15. There are many other banks shows the pattern similar to Indian bank. The productivity suffers in 2013-14 where many banks have index less than one. The other two estimates i.e. efficiency change and technical change from 125

year to year is given in the Table 5.4.2. Table 5.4.2: Efficiency change & Technical change over years Efficiency change Bank Name Dena B. J&K B. South Ind.B Syndicate B. Ind.Overs. . Andhra B, Karnataka . UCO B. ICICI B. OrienB.Com DCB B. Indian B. IDBI B. SBI B. of Baroda Union B.Ind. HDFC B. Corp B. Kotak Mahn. B. of India Canara B. PNB RBL B. Federal B. Allahabad B. Axis B. Cntrl B. Ind. Yes B. Karur Vysya

Technical Change

2012-13 2013-14 2014-15 2015-16 Average 2012-13 2013-14 2014-15 2015-16 Average 1.029 1.056 1.042 0.918 1.011 1.060 1.010 0.988 1.194 1.063 1.153 1.000 1.000 1.000 1.038 1.330 0.795 1.026 1.000 1.038 0.802 1.000 1.000 1.000 0.951 0.643 0.793 1.184 1.000 0.905 0.991 0.975 1.041 0.921 0.982 0.981 1.022 0.970 1.186 1.040 1.066 0.947 1.028 1.008 1.012 1.135 0.966 0.936 1.085 1.031 1.034 0.953 1.098 1.000 1.021 1.069 0.970 0.934 1.000 0.993 1.021 0.998 1.097 1.000 1.029 1.043 0.954 1.089 1.000 1.022 1.079 0.956 1.046 0.839 0.980 1.165 0.974 0.983 1.300 1.106 0.978 0.966 1.035 1.000 0.995 0.957 1.025 0.997 1.000 0.995 0.998 0.997 1.019 0.997 1.003 0.995 0.967 1.001 1.058 1.005 1.032 1.070 1.000 1.000 1.026 1.065 1.034 1.001 1.000 1.025 1.103 1.000 1.000 1.000 1.026 1.217 0.873 1.244 1.000 1.084 1.048 0.974 1.017 0.970 1.002 1.098 0.986 0.936 1.092 1.028 1.008 0.966 1.034 0.989 0.999 1.016 1.018 0.961 1.055 1.013 0.967 1.023 1.000 1.000 0.998 0.935 1.026 0.999 1.000 0.990 1.031 0.920 1.102 0.962 1.004 1.063 1.013 0.927 1.102 1.026 1.016 1.000 1.000 1.000 1.004 1.033 1.017 0.983 1.000 1.008 1.007 1.066 0.986 0.960 1.005 1.015 1.062 0.950 1.079 1.027 1.011 1.000 1.000 1.000 1.003 1.021 1.027 0.899 1.000 0.987 1.070 0.994 1.061 0.942 1.017 1.144 1.011 0.966 1.140 1.065 1.023 1.009 1.057 0.952 1.010 1.047 0.996 0.946 1.199 1.047 1.030 0.956 1.062 1.000 1.012 1.060 1.012 0.961 1.000 1.008 1.060 1.058 1.182 0.934 1.059 1.124 0.975 0.878 1.352 1.082 1.020 0.969 1.107 0.938 1.009 1.040 1.016 0.973 1.102 1.033 1.020 1.026 1.071 0.926 1.011 1.040 0.986 0.988 1.122 1.034 1.005 1.027 1.026 1.000 1.015 1.009 0.969 0.989 1.000 0.992 1.059 0.921 1.037 1.076 1.023 1.121 0.994 0.907 1.021 1.011 1.073 1.154 1.068 0.995 1.073 1.151 0.918 0.935 1.085 1.022 1.027 1.029 1.029 1.000 1.021 1.055 1.004 1.010 1.000 1.017

Similar to Table 5.4.1, 5.4.2 is also aggregate of two tables. The first part of table summarizes efficiency change index, and second part technical change index. The results show that there is 3.6% of efficiency change from 2012 to 2016. The banks with increased efficiency change are: Yes bank (score 1.073)by 7.3%, RBL bank (1.059) by 5.9%, followed by J&K bank with 3.8% and so on. The banks with decreased efficiency include: South Indian bank (0.951) by -4.9%, UCO bank (0.98) by -2.0%, Syndicate bank (0.982) by -1.8% and so on. The second half of the Table 5.4.2 summarizes the Technical change over the five year time span, which increased from 2012-13 and decreased from 2013 to 2015. Overall technical change index is slightly decreased by 0.3%. The banks with in126

creased technical change index are: UCO bank (1.106) by 10.6%, Indian bank (1.084) by 8.4%, RBL bank (1.082) by 8.2% and so on. Similarly the banks with decline of technical change over the years are: South Indian bank (0.905) by -9.5%, Kotak Mahindra bank (0.987) by -1.3%, Bank of Baroda (0.99) by -1.0% and so on.

5.5

Conclusion

We analyzed the efficiency of various Indian commercial banks in two ways. The first one consists in employing usual DEA to the specific year of 2016 to monitor the recent performance of the banks. Whereas, the second way consists in employing DEA-based MPI to the same banks for the five year span between 2012 and 2016. Our results indicate that J&K Bank, South Indian Bank, Andhra Bank, Karnataka Bank, ICICI Bank, DCB Bank, Indian Bank, Bank of Baroda, HDFC Bank, Kotak Mahindra Bank, PNB, Axis Bank and Karur Vysya Bank are considered to be benchmarks. These banks were efficient in all spheres and hence referred as MPSS (Most Productive Scale Size) DMUs. Further, in super efficiency context, South Indian bank is considered to be most efficient in all the years except 2014 where J&K overtakes it by a small margin. Also, in the context of MPI, we observed that total productivity change of a bank was more determined by technical change rather than efficiency change. The reason behind high technical change may be with the influence of external factors, such as RBI and government policies, whereas the efficiency change may be influenced by the operations of banks itself. In this study, the results show that the overall productivity of banks is increased by 2.4% from 2012 to 2016. This percentage is slightly low and mainly due to technical change. It is evitable from the results that there is more room for improvement of productivity through managerial practices.

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Chapter 6

Summary, Findings and Scope for Future Research 6.1

Introduction

Evaluation of relative efficiency or benchmarking of an entity is a systematic process of comparison among the entities one against each other. More generally, it is the comparison of performances of the production entities which utilizes the same type of resources to produce same type goods and services. These can be banks, finance companies, educational institutions, hospitals, production-firms, airports, seaports and so on. For the purpose of convenience in understanding, we named them as DMUs. The measurement of the performance of the DMUs concerned is an important improvement tool for staying competitive in the complex market situations. This helps in identification and exploration of limitations of the DMUs under study so that subsequent improvements could be made in order to survive and prosper in an environment of facing global competition. Measuring the performance indicators of a DMU looks easy for single input single output but it is difficult when the DMU possess several inputs and several outputs and even more difficult when the relationship between the inputs and outputs is complicated. However, it is a wellknown fact that the incorporation of too many variables can lead the DEA models to discriminate the efficiency scores insufficiently. Therefore, choosing of appropriate DEA-model with appropriate orientation is an important aspect. This research aims 128

to contribute evaluation procedures with the applications of Operational Research more specifically to the theory and methodology of Data Envelopment Analysis. It will be used for efficiency evaluation and benchmarking the DMUs under the context of several inputs and several outputs. DEA uses mathematical programming technique to monitor the performances of DMUs in relation to each other. The technique was proposed by Charnes et al. (1978) and immediately was recognized as an fantabulous methodology for examining the performance in different organizations. Additionally, its empirical orientation with no prior assumptions has resulted in its use in a huge number of studies involving about all profit and non-profit organizations. The CCR model got near about thirty thousand citations within three decades, which is enough to discuss its importance.

6.2

Summary and Contribution

This research study has made an attempt to focus on some contributions for performance evaluation in the area of DEA with orientations of both classical and empirical. As a part of contributions to the classical methods, we proposed some theoretical models to be employed in the resembled environments with certain assumptions. In the context of empirical contribution, certain models were employed. In the context of empirical contribution, some models were employed to the illustrative data-sets and exploration of possible inferences related to their performance and efficiency analysis. The core study has organized in presenting all the contributions in the four chapters of the thesis. Chapter one is an introductory chapter on performance evaluation through DEA approach. This chapter presents a brief introduction of all the methods of performance evaluation along with their respective advantages as well as disadvantages. As our research is based on the most important method (DEA) among all the methods of performance evaluation, all the related models were elaborated chronologically. The notion and methodology of DEA, basic DEA models like CCR and BCC with both input and output orientations were explained in Multiplier and additive forms. This also includes a brief introduction on two-stage DEA models, Double frontier DEA models, DEA window analysis and DEA-based Malmquist Productivity Index, etc. Further, the literature review of all these techniques was described. This

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chapter also provided the information on Research Gap and Motivation, Objectives, Data Acquisition Methodology and Organization of Thesis. The second chapter is primarily devoted to two-stage DEA processes. Twostage DEA processes are special cases of DEA process where the production passes through two different stages. Conventional DEA models deal with the evaluation of the efficiency by considering DMUs as black-box, where the internal structure of the production process is ignored and hence it lacks in identifying the sources of inefficiency. There are numerous real-life situations where production process passes through several internal stages and performance of these internal stages determine the standard/ quality of overall performance. For such cases, any evaluation method that ignores these internal stages in performance evaluation may result in inaccurate efficiency evaluation. The internal stages of the processes are either connected in series or in parallel. We concerned our study to the series type connection in twostage production processes. It is observed that there are four possible types of two-stage DEA processes with series type connection. This chapter elaborates the proposed non-radial SBM models for all of the four types of production processes. The models were also supported with respective numerical examples. Further, the process was generalized to multi-stage production process with Q series type stages. The SBM model for the q th divisional stage as well as network model for all the stages is also presented in the chapter. The third chapter deals with double frontier DEA models. Double frontiers DEA model evaluates the efficiency of a DMU based on two opposite frontiers namely optimistic and pessimistic frontiers. All the basic DEA models like CCR, BCC or any other radial and non-radial models evaluate the efficiency of DMUs based on only the optimistic frontier. The optimistic frontier assumes the favorable conditions for the production process. However, in real life situations, DMUs may face unfavorable conditions sometimes. Hence, applying the optimistic models for those types of situations is far from reality. Several studies were mentioned in this chapter which evaluated the DMUs based on pessimistic frontiers. Thus, if the nature of the production process is known optimistic and pessimistic models can be employed accordingly. In case, if there is any ambiguity regarding the conditions of favorability and non favorability in the production process then both the optimistic and pessimistic DEA models can be employed simultaneously. These two models represent the two opposite frontiers for a set of DMUs and all the DMUs lie within the convex set. Subsequently, two types of efficiency scores are computed and the overall 130

measure is calculated based on their geometric average. This chapter presents the proposed SBM double frontier DEA models for two-stage production processes. This includes optimistic and pessimistic models for the divisional stages as well as for the network processes. For examining the overall efficiency of the system we choose the method proposed by Wang and Chin (2008) rather than geometric average, because of its advantages over the geometric average. Furthermore, the chapter includes a numerical example of Taiwanese non-life insurance companies (Published Data set) to support our model. Chapter four is devoted to present DEA window analysis approach. This is the approach used to evaluate the efficiency change over time while keeping technology a constant. The models in the earlier chapters are limited for the particular period of time and failed to apply them in the time-varying and cross-sectional data. To prosper in the global competition, a DMU has to evaluate its performance from time to time. Since an efficient DMU at a particular time need not be efficient at any other time. Decision makers will be interested in evaluating the efficiency change between two time periods. DEA window analysis is that which will evaluate the efficiency change between any two time periods. This approach evaluates the performance of DMUs over time by treating them as different DMUs in each time period. In doing so, the performance of a DMU at any time period can be contrasted with its own performance at different time periods as well as to the performances of other DMUs in the study period. Thus, the number of data points will increase accordingly and will be more useful in small sample cases. This number is inversely proportional to the window width, i.e., lesser the window width will accommodate the more the data points and similarly the greater the window width will accommodate the lesser the number of data points. The window width is referred with the number of time periods included in the analysis in a particular window. The range of window width is somewhere between one and all the periods in the study horizon. The range of window width remains constant all over the windows. The underlying assumption of this approach is that it is working on the principle of moving-averages, i.e., one year is included in the analysis set while the other is withdrawn. This chapter also aims at presenting the window analysis approach in non-radial SBM form. The approach is supported with the numerical illustration of top ten Indian Cement Companies. We also evaluated the panel data through average SBM approach, which consider the averages of inputs and outputs throughout the span of the study period. The estimate of such efficiency scores provides a rough idea of how the DMU 131

had performed during the period of time. Apart from its biased efficiency measure, it provides a rough or crude measure of efficiency over the period of time. Chapter five is based on the DEA-based Malmquist Productivity Index (MPI). It is an index used to estimate the productivity change over time. As mentioned in chapter four, window analysis is used to evaluate the productivity change over time. The problem with window analysis is due to that it assumes the constant technology over time, and hence considers any change in productivity only due to managerial effects. However, productivity is very much affected by technological changes and so should be isolated from the efficiency change to have good managerial decisions. DEA-based MPI is the method used to differentiate the total productivity change into two mutually exclusive and exhaustive components such as technical efficiency change and technological change. The technical efficiency change is defined as the score related to best practice frontier (catching up), whereas, the technological change (innovation) is defined as the shift of best practice frontier from the time t to t+1. This chapter also described the input-oriented distance functions to evaluate the two components. The distance functions are based on the base technology (t) and current technology (t+1) for both the two periods, hence comprises of four distance functions. The inverse of the Farrell’s technical efficiency measures are equivalent to the input oriented distance functions. Thus, we present four equivalent LPP’s for the four different input oriented distance functions. The methodology for obtaining the scale efficiency change is also included in the chapter. An empirical illustration of DEA-based MPI on Indian commercial banks also envelops in the chapter.

6.3

Findings and Observations of the study

This section presents the major findings of our research study. We have proposed SBM models for four types of two-stage DEA processes. We observed that network models are more reliable when the production exhibits in number of internal stages. Network models helps in locating the source of inefficiency and that source will be accordingly rectified in such a way that the other factors will not be affected. We generalized the series type structure to multistage process with ’Q’ stages. Specific and particular network model can be explored from the generalized one with specific stipulations. For example, four stage network model can be obtained by substituting Q=4 in the generalized model. We have applied real time data on two developed

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models (Type-1 and Type-3) among the four proposed. Type-1 network model was applied on 27 Iranian Resin manufacturing companies. In the detailed view of the data finding and analysis, it is observed that most of the DMUs were inefficient either in stage one or in stage two. The DMUs namely Peka Chemie Co. and Peik Chimie Co. were found to be efficient in both stages and subsequently network efficient DMUs resembled with the definition of network efficiency. We also evaluated the efficiency while neglecting the internal processes as conventional models. It is observed that the efficiency estimates were unable to find inefficiency measures. Type-3 network model was applied on regional R&D processes of 30 provincial level regions in china studied with the numerical illustration used by Li et al. (2012). The details of analysis revealed that two DMUs namely Guangdong and Xinjiang were found to be efficient in both divisional stages and network processes. All other 28 DMU are inefficient either in divisional stage -1 or in divisional stage-2. Further, we proposed SBM models for two-stage production structure through optimistic and pessimistic perspectives presented in chapter three. The optimistic and pessimistic frontiers consists the two opposites frontiers of a set of DMUs. We observed that the double frontier based performance evaluation is more feasible if the conditions of the production processes are complex to identify. The overall efficiency score based on double frontiers for network process was evaluated using the expression proposed by Wang and Chin (2009) rather than the geometric average of two. We applied our model to the published data-set of Taiwan’s non-life insurance companies and estimated the optimistic and pessimistic efficiency scores for first stage, second stage and for network processes of all companies. Furthermore, we evaluated an overall efficiency score based on both these extreme and opposite efficiency scores which enable us to rank the companies according to their performance. The insurance company ’Asia’ got the first rank based on the overall efficiency score followed by Cathay Century, Fubon, Union and so on. On the other hand Mitsui Sumitomo was ranked last in terms of overall efficiency scores. As mentioned in chapter four, DEA window analysis is useful technique for the measurement of productivity change over the period of time. As non-radial models are non-orientation DEA models, we described DEA window analysis in SBM form. Further, we applied it on the numerical data-set of top ten Indian cement companies for the measurement of productivity change during the period 2007 to 2016. For compatibility and analysis purpose, we have chosen the window width as three 133

years. The SBM efficiency scores for 10 cement companies in 8 windows with width of 3 each are described in this chapter. It is observed that there is an increasing trend in the productivity change over the study period except in 2016 where it is slightly decreased. The Ultra-tech Cements was the leading company in the set of companies under study. Its SBM efficiencies in the first window are 100%, 74% and 68.5% respectively for the years 2007, 2008 and 2009. Subsequently, in the second window, the efficiency scores are 78.6%, 71.9% and 70.7% correspond to years 2008, 2009 and 2010 respectively. On the similar lines of observations, the table bestows itself to a study of ’trends’ and interrogation of ”stability” of efficiency scores. Two separate trends are revealed within windows by the adoption ’column views’ and ’row views’. Observing the scores of the ”Ultratech” Cement Company again in the second window, the efficiency score varies from 78% to 70% over the years 2008 to 2010 by adopting of ’row view’ perspective. On the other hand, its efficiency also varies among the windows by adopting ’column view’ perspective. This variation excogitates simultaneously both the absolute performance of a cement company over time and relative performance of that company in comparison to its peers. In contrast, our findings reveal that ’OCL India’ was the worst performer with 50.1% mean SBM efficiency with a standard deviation of 0.175. We noticed from the results that all the companies exhibited betterment and upward trend during the later part of studies. The results of DEA-based Malmquist Productivity Index are presented in the chapter five of the thesis. Herein, we obtained productivity change, efficiency change and technical change of various Indian commercial banks from 2012-16 through DEA based MPI. This technique not only evaluates the productivity change as evaluated by window analysis, but also decomposes it into efficiency change and technological change. We analyzed the efficiency of various Indian commercial banks for both a specific year 2016 and five year span between 2012 and 2016 through DEA and Malmquist Productivity Index respectively. In case of DEA analysis, we evaluated CCR and BCC efficiency scores in both input and output orientations along with scale efficiency for the year 2016 because of its recentness. The results reveals that J&K Bank, South Indian Bank, Andhra Bank, Karnataka Bank, ICICI Bank, DCB Bank, Indian Bank, Bank of Baroda, HDFC Bank, Kotak Mahindra Bank, PNB, Axis Bank and Karur Vysya Bank are considered to be benchmarking DMUs with respect to their MPSS (Most Productive Scale Size), technical efficiency and scale efficiency. Further, for ranking purpose, super efficiency scores were evaluated. 134

South Indian bank got the topmost rank in all the years except 2014 where J&K overtakes it by a small margin. Also, in the context of DEA-based MPI, we found that total productivity change of a bank was influenced heavily by technical change rather than efficiency change. The reasons behind high technical change may be with the influence of external factors, such as RBI and government policies, whereas the efficiency change may be influenced by the operations of banks itself. In this study, the results show that the overall productivity of banks is increased by 2.4% from 2012 to 2016. This percentage is slightly low and mainly due to technical change. It is evitable from the results that there is more room for improvement of productivity through managerial practices.

6.4

Scope for Future Research

As this work is more significant in performance evaluation of decision making units through non-parametric approach, there is a good scope for these sort of studies in the contexts of management Decision Making. There are numerous issues in theoretical and empirical perspectives that may be looked into for the further research and few among them are stipulated as follows: • In chapter two, our study was limited not only to series type DEA process but also limited to two stages. The number of stages can be increased to more than two stages where the internal stages are connected either in parallel-type or series-type or even combination of both. • The problem with two-stage DEA models is that is assigns higher values (observed from our analysis and literature review) to first stage than to second stage. Since in first stage if we maximize the outputs which are used as inputs in second stage, first stage will maximize the efficiency to maximum which results in the low efficiency score for the second one. One can develop models to fix the efficiency so that divisional stages are not discriminated. Further, these models can be empirically employed in various other dimensions • In the case of multi-stage production processes, all the divisional efficiency scores can be evaluated simultaneously without any discrimination through multi-objective programming problems. 135

• All the DEA models can represented in optimistic and pessimistic way. Thus models either in radial as well as in non-radial forms can be presented in double frontier form and an overall measure can be evaluated either through geometric average or the expression used in chapter-3. Additionally, our model can be employed empirically in any other dimension. • DEA window analysis can be employed on numerical data sets to examine the efficiency fluctuations for various profit and non-profit organizations. Further, it can also be developed in two-stage and multi-stage networking processes. • Evaluating the efficiency trend through DEA window analysis leads to prediction of future trends and sensitivity analysis can be done accordingly to obtain the limits to remain efficient in the future. • As DEA-based MPI is the empirical contribution to our study, it can be employed also in various other dimensions. Further, MPI can be proposed in Free Disposal Hull (FDH) approach to obtain integer solutions. • DEA-based MPI can also be extended to two-stage and multi-stage production processes. Further, DEA-based MPI can be employed in cases of undesirable outputs. • All the proposed models can also be presented in fuzzy environment to deal with the fuzziness in the data-sets.

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List of papers published / communicated Papers published (1) Arif Muhammad Tali, Tirupathi Rao Padi, and Dar, Qaiser Farooq (2017), “Slack- based Measures of Efficiency in Two-stage Process: An Approach Based on Data Envelopment Analysis with Double Frontiers”, International Journal of Latest Trends in Finance and Economics Sciences, Vol. 6(3), pp: 1194-1204. (2) Arif Muhammad, Tirupathi Rao, and Qaiser Farooq (2017), “ DEA Window Analysis with slack-based measure of Efficiency in Indian Cement Industry”, Statistics, Optimization & Information Computing.(SOIC). (Accepted) (3) Dar, Qaiser Farooq, Tirupathi Rao Padi, and Arif Muhammad Tali (2016), “Mixed input and output orientations of Data Envelopment Analysis with Linear Fractional Programming and Least Distance Measures”, Statistics, Optimization and Information Computing , Vol. 4(4), pp: 326–341. (4) Dar, Qaiser Farooq, Tirupathi Rao Padi, and Arif Muhammad Tali (2017), ”Decision Support System through Data Envelopment Analysis & Stochastic Frontier Analysis”, International Journal of Modern Mathematical Sciences, Vol. 15(1), pp: 1-13. (5) Arif Muhammad Tali, Tirupathi Rao Padi, and Dar, Qaiser Farooq (2017), “DEA Double Frontier in Two-Stage Processes with Slack-Based Measure”, International Journal of Modern Management Sciences, Vol. 5(1), pp: 23-36.

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Papers Communicated (1) Arif Muhammad, Tirupathi Rao, and Qaiser Farooq, “Average Slack-Based Measure of efficiency in DEA: An Approach to Indian Commercial Banks ”, Communicated to International Journal of Mathematics and Mathematical Sciences (2017). (2) Arif Muhammad Tali, Tirupathi Rao Padi, and Dar, Qaiser Farooq, “Nonradial DEA models in Series Type Two-Stage Production processes”, Communicated to OPSEARCH (2017) (3) Arif Muhammad Tali, Tirupathi Rao Padi, and Dar, Qaiser Farooq, “MultiPeriod Performance Evaluation of Indian Commercial Banks Through Data Envelopment Analysis and Malmquist Productivity Index ”, Communicated to Journal of Decision Systems (2017) (4) Dar, Qaiser Farooq, Tirupathi Rao Padi, and Arif Muhammad Tali, “Data Envelopment Analysis with Sensitivity Analysis and Super-Efficiency in Indian Banking Sector ”, Communicated to SORT- Statistics and Operations Research Transaction (2017). (5) Dar, Qaiser Farooq, Tirupathi Rao Padi, Arif Muhammad Tali and Aasif Shah, “Data Envelopment Analysis in Three-Stage Networking Structure with Slack Based Measure (SBM) ”, Communicated to Elsevier Journal Measurement. (6) Dar, Qaiser Farooq, Arif Muhammad Tali, Tirupathi Rao Padi and Hamid, Y “Fuzzy Data envelopment Analysis with SBM using α−level Fuzzy Approach”, Communicated to Pakistan Journal of Statistics and Operation Research (2017). (7) Aasif Shah, Arif Muhammad and Qaiser Farooq “Multi-Scale Beta Estimation through Wavelets”, Communicated to Financial Innovation (2017).

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Appendix

Appendix-I Charnes Cooper Transformation of Radial Models Suppose the fractional programming problem for evaluating the radial efficiency score Ejo where jo is the DMU under evaluation is as follows; Ps ur yrjo M aximize Ejo = Pr=1 m i=1 vi xijo

(i)

Subject to Constraints; Ps ur yrjo Pr=1 ≤ 1; ∀ j = 1, · · · , n m i=1 vi xijo ur , vi ≥ ε; ∀r, i Here Ejo is the efficiency score of DMU jo with ur and vi as the respective weights of rth output and ith input respectively. The ε is a non-Archimedean value to ensure strict the positivity of weights of all inputs and outputs. The objective function consists the ratio of two linear functions and hence called as linear Fractional Problem. The Problem with this type of models are that of their complex computations. Additionally, solving these types models often gives multiple optimal solutions. Thus, it looks important to transform the fractional form into linear form through the popular methodology of Charnes Cooper Transformation. There are two types of normalization’s, one is numerator normalization (Output oriented) and the other is denominator normalization (Input oriented). The denominator normalization proceeds as follows; Suppose 1 i=1 νi xijjo

t = Pm Substituting this in model (i), we have;

152

M aximize Ejo =

s X

t ∗ ur yrjo

(ii)

r=1

Subject to Constraints; m X t ∗ vi xijo = 1 i=1 n X

ur yrjo −

j=1

n X

vi xijo ≤ 0; ∀ j = 1, · · · , n

j=1

ur , vi ≥ ε; ∀r, i Now substitute t ∗ u r = µr

M aximize Ejo =

s X

and

t ∗ vi = νi

...in (ii)

µr yrjo

(iii)

r=1

Subject to Constraints; m X νi xijo = 1 i=1 n X

ur yrjo −

j=1

n X

vi xijo ≤ 0; ∀ j = 1, · · · , n

j=1

µr , νi ≥ ε; ∀r, i In a similar way numerator normalization involves the following substitutions in model (i); 1 and t ∗ ur = µr , t ∗ vi = νi r=1 ur yrjjo

t = Ps

153

We get the following output oriented model; M inimize Ejo =

m X

νi xijo

(iv)

i=1

Subject to Constraints; s X µr yrjo = 1 r=1 n X

ur yrjo −

j=1

n X

vi xijo ≤ 0; ∀ j = 1, · · · , n

j=1

µr , νi ≥ ε; ∀r, i Appendix-II Charnes Cooper Transformation of Non-Radial Models Consider the fractional form of SBM models as;

M inimize τj0 =

1−

1 m

1+

1 s

Si− i=1 xij0

Pm

Sr+ r=1 yrj0

Ps

Subject to Constraints; n X Λj xij + Si− = xij0 ; i = 1 , 2 , 3 , . . . m. j=1 n X

(v)

Λj yrj − Sr+ = yrj0 ; r = 1 , 2 , 3 , . . . s.

j=1

Λj ≥ 0; Si− ≥ 0; Sr+ ≥ 0, ∀j, i, r. Since SBM models are non-orientations, both normalization’s leads to same thing. Let us substitute

t=

1 P 1 s Sr+ 1+ s r=1 xijjo

154

which leads the model (v) to; m

M inimize τj0 = (t ∗ 1) −

S− 1X t∗ i m i=1 xij0

Subject to Constraints; s 1X Sr+ (t ∗ 1) + t∗ =1 s r=1 yrj0 n X

(v)

t ∗ Λj xij + t ∗ Si− = t ∗ xij0 ; i = 1 , 2 , 3 , . . . m.

j=1 n X

t ∗ Λj yrj − t ∗ Sr+ = t ∗ yrj0 ; r = 1 , 2 , 3 , . . . s.

j=1

t ∗ Λj ≥ 0; t ∗ Si− ≥ 0; t ∗ Sr+ ≥ 0, ∀j, i, r. and t > 0 + + On substituting τ = ρ; t ∗ Si− = s− i ; t ∗ S r = sr ;

t ∗ Λj = λj , we have;

m

M inimize ρj0 = t −

1 X s− i m i=1 xij0

Subject to Constraints; s 1 X s+ r t+ =1 s r=1 yrj0 n X

(vi)

λj xij + s− i = txij0 ; i = 1 , 2 , 3 , . . . m.

j=1 n X

λj yrj − s+ r = tyrj0 ; r = 1 , 2 , 3 , . . . s.

j=1 + λj ≥ 0; s− i ≥ 0; sr ≥ 0, ∀j, i, r. and t > 0

Which is the required SBM model. Thus, every model in fractional from irrespective of approach can be transformed through Charnes Cooper Transformation.

155